# Into the Single Cell Multiverse:
an End-to-End Dataset for Procedural Knowledge Extraction in Biomedical Texts

Ruth Dannenfelser1
Jeffrey Zhong1
Ran Zhang2
Vicky Yao1

1 Department of Computer Science, Rice University
2 Department of Genome Sciences, University of Washington Address
correspondence to<EMAIL_ADDRESS>
###### Abstract

Many of the most commonly explored natural language processing (NLP)
information extraction tasks can be thought of as evaluations of declarative
knowledge, or fact-based information extraction. Procedural knowledge
extraction, i.e., breaking down a described process into a series of steps,
has received much less attention, perhaps in part due to the lack of
structured datasets that capture the knowledge extraction process from end-to-
end. To address this unmet need, we present FlaMBé (Flow annotations for
Multiverse Biological entities), a collection of expert-curated datasets
across a series of complementary tasks that capture procedural knowledge in
biomedical texts. This dataset is inspired by the observation that one
ubiquitous source of procedural knowledge that is described as unstructured
text is within academic papers describing their methodology. The workflows
annotated in FlaMBé are from texts in the burgeoning field of single cell
research, a research area that has become notorious for the number of software
tools and complexity of workflows used. Additionally, FlaMBé provides, to our
knowledge, the largest manually curated named entity recognition (NER) and
disambiguation (NED) datasets for tissue/cell type, a fundamental biological
entity that is critical for knowledge extraction in the biomedical research
domain. Beyond providing a valuable dataset to enable further development of
NLP models for procedural knowledge extraction, automating the process of
workflow mining also has important implications for advancing reproducibility
in biomedical research.

##  1 Introduction

The recent onslaught of pre-trained language models has spurred on tremendous
advances in a range of natural language processing (NLP) applications,
including named entity recognition (NER), named entity disambiguation (NED),
sentiment analysis, and relation extraction [1, 2, 3, 4, 5]. These
applications mostly fall under the umbrella of tasks that aim to extract
declarative knowledge, sometimes also referred to as “knowing that,” since
these tasks focus on matters of factual knowledge (e.g., knowing that “neuron”
is a cell type) [6, 7]. Declarative knowledge is often contrasted with
procedural knowledge, or “knowing how,” (e.g., knowing how to conduct an
experiment) [6, 7]. Early AI researchers raised the importance of developing
representations of procedural knowledge, given that performing plans or
procedures is a fundamental way in which humans navigate the world [8].
However, compared with declarative knowledge extraction, there remains a vast
gap in the development and application of machine learning methods towards
procedural knowledge tasks [9].

Recently, there has begun to be a renewed interest in using machine learning
to model procedural knowledge, especially knowledge extraction from text using
NLP. These efforts have mostly focused on cooking and other common household
tasks [10, 11], business processes [12], and technical manuals or
manufacturing [13]. The specific applications that have garnered interest seem
to have been naturally motivated by either the emergence of valuable datasets
(e.g., online recipes for cooking, WikiHow for various how-to tasks) or
economic gain through business process optimization. Interestingly, one of the
main ways scientists and engineers communicate their findings—through academic
papers—is a prime source of unstructured text describing “know-how,” yet few
studies explore extracting procedural knowledge from scientific literature.
This is the case though there is also an abundance of open access scientific
literature that is frequently used for many standard declarative knowledge
extraction studies.

We posit that there are 3 main reasons that procedural knowledge extraction
from scientific literature is not currently widely studied:

  1. 1.

Though most research papers will describe procedures, i.e., methods, they are
typically not written with as much structure as a recipe or technical manual,
and thus not as easy to model “off the shelf.” In fact, methods sections are
often organized by thematic categories and do not necessarily represent the
“temporal ordering” in which the individual steps were done.111Note that here,
temporal ordering is used loosely, as we are simply referring to the workflow
ordering that a reader can deduce from the manuscript. It is of course common
that scientific manuscripts present their main results in differing order than
originally conducted. That said, we expect that internal ordering of tasks
within each major result to be typically a good reflection of what was
actually performed. It is also often the case that the results sections need
to be read together with the methods sections to reconstitute how various
tools were used.

  2. 2.

There can be varying degrees of ambiguity in a scientific manuscript when
systematically describing a workflow. The same method or software tool can be
used at several time points throughout a paper, but in different contexts and
for different purposes. For example, principal component analysis (PCA) can be
used for dimensionality reduction, feature selection, or visualization.
Failure to account for context may lead to a workflow that appears to simply
have a chain of PCAs. In addition, multiple parallel workflows can be
described in a single paper. For example, a single paper can consider multiple
datasets, each of which are processed differently, before they are analyzed
jointly.

  3. 3.

Unlike writing down recipes or household tasks, annotating the workflow used
in a scientific paper is challenging without domain expertise, thus resulting
in a bottleneck for developing structured datasets.

Motivated by these observations, we introduce FlaMBé (Flow annotations for
Multiverse Biological entities). FlaMBé is a collection of structured
annotations in biomedical research papers, with a particular focus on
computational analysis pipelines in single cell research. While scientists
have long been interested in studying single cells [14, 15], it was with the
introduction of high-throughput single cell sequencing technologies around
2010 [16] that this area has exploded in activity, not only in applications of
this experimental technique to various biomedical applications, but also in
the development of computational tools and software to analyze the resulting
data. Recent efforts to wrangle the space of analysis tools has resulted in
specialized databases such as scRNA-tools [17], which currently tracks over
1,500 software tools across over 30 analysis tasks.222The terminology scRNA-
tools uses for these analysis tasks is “categories,” since they are focused on
grouping tools by their applications. We simplify the terminology here to make
clear that each tool can have multiple category tags. Interestingly, the
majority of tools catalogued by scRNA-tools are used for more than one
analysis task, and one of the most commonly used tools, Seurat [18], is
associated with as many as 10 categories of tasks, further highlighting the
importance of considering context.

Figure 1: Example overview of the workflow of tool contexts (analysis tasks).
Summary figure of workflows from different tool contexts captured by FlaMBé.
Direction of edge represents which analysis task was completed prior to the
output being sent to the following task. Weight of edges represent the number
of papers that mentioned. Node size corresponds to degree, i.e., the number of
papers that mentioned the corresponding analysis task.

In FlaMBé, we develop a structured representation of the procedural knowledge
represented in scientific literature by considering (1) the targets of the
study, which in the case of single cell research, are the tissues and/or cell
types that are assayed; (2) the tools applied in the study as well as the
analysis task or context in which they are being used; and (3) the workflow
between tools and analysis tasks, e.g., when PCA is used for dimensionality
reduction before the results are clustered using DBSCAN. Part of the
motivation in structuring FlaMBé in this manner is that we can break down the
more complex, unstructured goal of procedural knowledge extraction into
existing, more manageable declarative knowledge extraction tasks. For example,
the identification of targets and tools in text reduces to NER and NED tasks.

Overall, we present 55 full text papers, including over 420,000 tokens,
annotated for relevant entities and relations from the PubMed Central Open
Access Subset by domain experts (computational biologists). To improve
coverage over a more diverse set of journals and entities, we also provide
tissue/cell type annotations in 1,195 paper abstracts mined from PubMed,
covering over 270,000 tokens. The entire dataset provides entity annotations
as well as disambiguation, where entities are linked to identifiers in
relevant knowledge bases. To our knowledge, FlaMBé is the largest NER and NED
dataset for tissues/cell types. Furthermore, we also provide annotations for
software tools and computational methods, also capturing 28 unique contexts in
which the tools are used for single cell research and nearly 400 workflow
relations between (tool, context) pairs. An example visualization of the flow
between contexts is shown in Fig. 1. FlaMBé is available for exploration and
download at https://github.com/ylaboratory/flambe.

We illustrate some example use cases for FlaMBé here, but the richness of this
dataset has many more potential downstream applications in machine learning as
well as computational biology and the wider biomedical field. In general, the
complexity of working with single cell data and its capacity for a variety of
different workflows, together with its important biomedical applications, is
ultimately what led us to choose the area of single cell research for FlaMBé.
However, we have also proposed a systematic framework to distill procedural
knowledge into a structured dataset in a manner that considers some of the
unique challenges of scientific literature. It is our hope that FlaMBé
provides a useful foundation for future “science-know-how” modeling and
datasets.

##  2 Related Work

FlaMBé is designed to represent a collection of complementary tasks that
together form the basis of a structured representation that captures
procedural knowledge in biomedical texts. Here, we discuss related datasets
and research efforts.

#### Biomedical NLP

Systematic evaluations of language models in a variety of different
benchmarking efforts have revealed that for specialized domains like
biomedicine, language models developed using domain-specific text (e.g.,
scientific literature) often outperform general-domain language models (e.g.,
trained on Wikipedia, news articles, webpages, etc.) on domain-specific tasks
[19, 20, 21, 22]. Furthermore, it seems that mixed-domain pretraining can
sometimes hurt more than help, suggesting that transfer learning is at times
unsuccessful due to how different general-domain text is from biomedical text
[20]. In general, there has been a demonstrated need for both domain-specific
pretrained language models as well as domain-specific datasets for method
benchmarking.

In the biomedical domain, language models are often trained on a mix of
abstracts from PubMed and full text articles from PubMed Central [21, 23, 19,
20], at times also with additional scientific text such as medical records
[24]. A variety of biomedical NLP benchmarking datasets have also been
developed [24, 20, 25], but often individual tasks can be fragmented. Very
recently, large scale efforts like BigBio [26] have systematically organized
comprehensive public collections of biomedical NLP datasets. BigBio’s curation
revealed that the largest represented task within biomedical NLP is
unsurprisingly NER, as there are a variety of biological entities that are
often of interest for text mining (e.g., diseases, gene names, chemical
compounds, anatomy/tissue/cell type). Of particular relevance to our work here
are previous dataset curation efforts for tissue/cell type [27, 28, 29, 30].
However, not only are these datasets smaller in terms of total annotations in
comparison with FlaMBé, but furthermore, none provide NED. Disambiguating
these terms and linking them to a systematic knowledge base provides more
utility for the biomedical community and also enables incorporation of
information from the associated knowledge base for improved knowledge
extraction.

The other entity that has recently begun to be considered for NER in
biomedical literature is software. As the field of computational biology grows
and, accordingly, the number of software tools and computational methods,
systematic identification and analysis of tool usage has become more relevant.
Large-scale curation efforts for NER and NED here include bioNerDS [31],
SoftCite [32], and SoMeSci [33]. These previous datasets have differing
limitations. Both bioNerDS and SoftCite only consider articles published
before 2011333bioNerDS further restricts its annotations to only two journals,
BMC Bioinformatics and Genome Biology. in their dataset, while SoMeSci curates
articles as recent as 2020. However, SoMeSci’s main endpoint is a knowledge
graph and thus does not provide its annotations in an easily usable format.
Both SoftCite and SoMeSci have been used as training data to automatically
identify software mentions across millions of scientific articles [34, 35],
though the resulting automatically annotated datasets differ greatly. Finally,
all previous datasets focus solely on software. Because one of the key goals
of FlaMBé is to extract data processing and analysis workflows, we also wanted
to expand annotations to computational methods that are often referred to in
scientific papers without necessarily a specific associated software (e.g.,
PCA, SVM).

#### Procedural knowledge extraction

Recent efforts in procedural knowledge extraction have been spurred on by the
increasing availability of naturally arising procedural knowledge-related data
sources. In fact, the widespread availability of online recipes have given
rise to the new research area of “food computing.” [10] Other areas where
there is active research in procedural knowledge extraction include household
tasks based on mining data sources such as WikiHow, Instructables, and eHow
[11], technical manuals [13], and business processes [12].

There has also been some limited attempts to examine scientific literature as
an application area. Song et al. propose representing procedural knowledge as
(target, action, method) triplets based on MEDLINE abstracts [36], and Halioui
et al. consider using process-oriented case-based reasoning to extract
workflows from papers mentioning phylogenetic analyses from PubMed [37].
Interestingly, these two pieces of work fall on two ends of the spectrum in
terms of the complexity of the representations they propose. In addition to
the limitations of modeling an entire workflow from only an abstract, Song et
al.’s proposed representation is also unable to take into account when tools
are applied in different contexts. Meanwhile, Halioui et al.’s representation
is somewhat arduous, and their contribution is mostly focused on a rule-based
workflow extraction framework rather than the assembly of a dataset that can
be used by other methods. Neither Song et al. nor Halioui et al.’s datasets
are accessible.444Halioui et al. provide a link to their data and framework
implementation in their paper, but the link is no longer active.

##  3 Dataset Collection Methodology and Overview

Annotations for NER, NED, and other knowledge extraction tasks were curated by
domain experts in computational biology for a series of 55 biomedical full-
text papers and 1,195 abstracts, indexed on PubMed Central (PMC) and PubMed,
respectively. We chose to include both full text and abstracts in FlaMBé to
have a breadth of unique tokens as well as the depth needed to extract
meaningful biological workflows.

###  3.1 Collection Methodology

#### Abstract corpus

The abstract corpus was hand-curated for tissue and cell type terms across 20
high-impact biomedical journals (full list in Supplementary Materials). To
ensure that no single journal was overrepresented due to publication quantity,
we set the number of sampled abstracts per journal to 60. Furthermore, we only
sampled from recently published works between 2016 and 2021, as advances in
technology have made it possible to study cell types in addition to bulk
tissue and we want to capture the new diversity of cell types in our
annotations. All abstracts were downloaded using PubMed eutils. To enable
evaluation of interannotator agreement (Supplementary Materials), each of 3
annotators was assigned 400 abstracts (60 from each unique journal), with 240
overlapping abstracts evenly distributed across journals.

#### Full text corpus

Because of the focus on single cell research, we used Pubmed eutils to query
PMC for 3 general article types (“Classical Article,” “Clinical Study,” and
“Journal Article”) using the following key words (allowing dashes to be used
as a connector as well): “scRNAseq,” “single cell RNAseq,” “single cell RNA
sequencing,” “single cell transcriptomics,” “single cell transcriptome.” Full
text articles were downloaded directly via the PMC FTP and parsed using Pubmed
Parser [38]. Out of the 55 total full text articles annotated by 2 annotators,
10 papers were annotated by both to evaluate interannotator agreement
(Supplementary Materials).

###  3.2 Annotation Types

Tissue, cell type, tool, and method were annotated using the Prodigy software
tool developed by Explosion AI for easy tracking of token-level tags. Due to
the more limited presence of tool and methods, ergo tool context and workflow
in abstracts, these annotations were only completed in the full text corpus.
Tissue and cell type were annotated in both the abstract and full text
corpora.

#### Tissue and cell type

To determine what classifies as tissue or cell type label, we use the terms in
the NCI Thesaurus,555https://ncithesaurus.nci.nih.gov/ncitbrowser/ a
comprehensive biomedical ontology for describing human samples which has
cross-references to many other biomedical ontologies, as a guide. We focus on
annotating useful sample descriptors that capture what biological entity is
being studied, and try to tag the most specific term possible (e.g., “left
ventricle” vs. “ventricle”). The full set of annotation rules given to each
annotator can be found in the Supplementary Materials.

A tissue or cell type in the text may be more specific than a term in the
ontology, or it may not match exactly or any of the given synonyms. In these
cases, we manually disambiguated the tag back to its nearest term in the
ontology. In all other cases we programmatically mapped exact matches and
synonyms back to NCIT identifiers. Additionally, in some cases, to express the
specificity found in the text, we used two terms from the ontology in the
disambiguation (e.g., “adipose stem cell” is mapped to two terms in NCIT
“adipose” and “stem cell“).

#### Tool and methods

Unlike tissues and cell types which have standardized ontologies, there is no
concrete vocabulary to annotate tools and methods in biomedical research. We
have done our best to define two concrete categories of methods, those where
an important computational transformation of the data has taken place but can
be done by more than one package, (e.g., K-means clustering or PCA), and those
that reference a specific tool or package. We label each of these respective
types as unspecified method (“UNS_METHOD”) or tool (“TOOL”). Furthermore, we
aimed to identify computational methods applied on data that are separate from
sequencing technologies and their related protocols (e.g., those done on
machines which physically handle a biological sample) and only annotate tools
and methods starting from the initial processing off of sequencing machines.

#### Tool context

In addition to annotating tools, methods, tissue, and cell type terms in the
full text we also provide a set of tool “contexts,” or the analysis task that
they are used for to process or augment data. This is important, as a single
tool may have multiple functions or reasons that it was applied (Fig. 3 shows
an example paper where Seurat used in 4 different contexts). For the sake of
exploring the single cell multiverse, we restricted the set of modes to
important functions in processing a wide variety of sequencing data. A single
mention of a tool in the text can have one or more modes assigned to it based
on its surrounding context. The full vocabulary for modes can be found in
Supplementary Materials.

#### Workflow

On a paper level, we aim to extract the various workflows done to samples,
where samples are defined as an assay (e.g., scRNA-seq, ChIP-seq, BS-seq, etc)
and a sample descriptor such as tissue/cell type pair. Once a unique set of
samples per paper are identified we link them with tool and mode pairs from
the text. Next, we annotate the flow by tabulating all edge pairs, where a
pair of tools with their corresponding modes are applied to a given sample. In
cases where an unspecified important transformation took place, such as an
‘UNS_METHOD‘ we use “unspecified_mode” as a placeholder. In this way we can
reconstruct and model multiple workflows in a paper when more than one sample
type is used.

###  3.3 Dataset description and statistics

#### Token level tags

All token level tags, such as those for tissue and cell type and tool and
method annotations are released as IOB and CoNLL files. The CoNLL files
contain disambiguated annotations, with the tissue and cell type tags mapped
semi-manually back to NCI Thesaurus identifiers and tools disambiguated back
to a standardized name. An additional description file is also provided, one
for tissues and cell types, which maps NCI Thesaurus ids to names, and one for
tool and method annotations, with annotations to relevant references, GitHub,
or project links.

Together, the full-text tag files span 55 papers and 429,373 tokens with 405
disambiguated (776 before disambiguation) tissue and cell type terms, 217
disambiguated tools, and 48 unique general methods. The abstract only tag
files span 1,195 papers with 272,771 tokens annotated and 286 disambiguated
tissue and cell type terms.

#### Tool context annotations

Mode annotations for the various tools are provided in the tool and method
CoNLL files. Each mode is manually assigned using the surrounding sentence
context.

#### Workflow annotations

There is no predefined standard format for paper-level knowledge extraction
annotations, so we split them into the following 3 files for easy parsing: A
sample description and identification file, containing a listing of unique
sample assay and tissue and cell type pairs; a tools applied file linking
samples with the tool-mode combinations covering modes; and tool sequence file
that ties pairs of tool-mode combinations together with sample identifiers.
These files cover 8 unique assays, across 28 tool modes, capturing 390 tool-
tool steps. There are on average, 10 workflow steps for each of the 38 papers
with a defined workflow.

##  4 FlaMBé Use Cases

The diverse collection of annotations in FlaMBé enables several different use
cases. We explore 3 example use cases of NER, tool context prediction, and
workflow visualization before discussing other potential downstream
applications.

#### Use case 1: named entity recognition

We illustrate how the IOB and CoNLL files can be used to train BERT models to
predict tissue and cell type mentions in biomedical abstracts. Using the full-
text data as training and our abstract annotations as the hold out set for
evaluation, we fine-tuned some of the most popular BERT models on HuggingFace
(Table 1) for NER prediction. All models perform reasonably well, with
PubMedBERT [20] having the best F1 for the cell type and tissue type
identification tasks. In general, the domain-specific pretrained language
models do tend to perform better than the general domain models, especially
when it comes to recall.

Table 1: Predictive performance (P/R/F1 scores) of various language models on abstract tissue/cell type annotations. Language models were fine-tuned on a combination of full text and abstracts and evaluated on a mixture of both text types for cell type and tissue annotations. Best performers are highlighted in bold. | Cell Type | Tissue
---|---|---
| Precision | Recall | F1 | Precision | Recall | F1
BERT-base [39] | 0.740 | 0.823 | 0.779 | 0.813 | 0.811 | 0.812
ELECTRA [40] | 0.775 | 0.802 | 0.788 | 0.819 | 0.841 | 0.830
BioBERT [21] | 0.725 | 0.806 | 0.763 | 0.848 | 0.859 | 0.854
BlueBERT [24] | 0.699 | 0.838 | 0.762 | 0.818 | 0.851 | 0.834
BioELECTRA [23] | 0.719 | 0.855 | 0.781 | 0.751 | 0.894 | 0.816
PubMedBERT [20] | 0.795 | 0.832 | 0.813 | 0.844 | 0.868 | 0.856

We also aim to demonstrate the utility of our annotations by comparing them
with the only other easily obtainable software annotation dataset, Softcite
[32], a resource that provides annotations of software mentions in full-text
research publications in the life sciences and economics. Here, we partition
FlaMBé’s full-text tool annotations into two sets of full-text data, holding
out 11 randomly chosen papers for evaluation. We use the remaining 44 papers
from FlaMBé and the entirety of Softcite for training. Both datasets were used
to train PubMedBERT, one of the consistent performers in tissue/cell type
prediction (Table 2). Despite being a smaller set of annotations, FlaMBé
outperforms Softcite, especially when it comes to identifying the full name of
a tool, (e.g., “Search Tool for the Retrieval of Interacting Genes/Proteins,”
more commonly known as “STRING”). This observation seems to be supported when
we examine the predictive performance broken down by tag type—the largest
performance difference between a model trained on Softcite and FlaMBé is in
the ‘I-Tool’ token (see Supplement). We hypothesize that the fact that
biomedical tools often have long, multi-word names (and corresponding acronym)
may play in role in this large difference. Of course, we note that in this
comparison FlaMBé has the advantage of using the same annotation criteria in
both the training and test sets, but nevertheless, we believe it still
illustrates the importance and utility of FlaMBé’s biomedical specific tool
annotations.

Table 2: Predictive performance (F1 scores) of PubMedBERT on tool annotations when using Softcite or FlaMBé (excluding papers used for evaluation) as training standard. Tool annotations from 11 full text papers were held out from FlaMBé as an evaluation standard. PubMedBERT was fine-tuned on either the entirety of Softcite annotations or the smaller FlaMBé training standard. | Precision | Recall | F1
---|---|---|---
Softcite [32] | 0.397 | 0.528 | 0.453
FlaMBé | 0.779 | 0.909 | 0.839

#### Use case 2: tool context prediction

As a proof of concept, we also used FlaMBé’s tool context annotations and
trained a PubMedBERT model to predict a tool’s context given the sentence in
which it is mentioned, akin to sentiment classification. We assembled a small
set of training (191 sentences over 28 papers) and test (45 sentences over 8
papers) data, limiting ourselves to sentences containing a mention of at least
one of the top 5 most mentioned tools, Seurat, Cellranger, t-SNE, Monocle, and
STAR, each of which can be applied in multiple contexts. We then trained
PubMedBERT models to predict context for each sentences in a one vs rest
framework, for contexts that are well represented in the test and training
datasets: Alignment, Marker Genes, and Clustering. Each of the classifers
performed well, with the alignment (AUC = 0.954) and marker gene (AUC = 0.953)
contexts being more distinguishable and clustering (AUC = 0.810) being the
most difficult. Given this promising performance on a test case, we anticipate
that more sophisticated methods will be able to achieve consistently strong
performance with our annotations.

#### Use case 3: visualization and exploration of different scientific
workflows

Different workflows can be extracted from FlaMBé’s annotations, at different
levels of specificity, either by highlighting the different tools used in a
paper (Fig. 2A) or the different tool contexts in a paper (Fig. 2B). These can
also be combined to extract more exact methodology (Fig. 3). Benchmarking
papers or work introducing a new tool have to compare with previous work and
create interesting workflows, as a small set of sample types is processed with
slight variations through different levels of an entire pipeline depending on
a paper’s objective (Fig. 2). Meanwhile, papers that seek to solve a
biological problem often have a more defined flow, with fewer tools from
sample to one or more endpoints (Fig. 3). By extracting these workflows, we
can not only classify the type of paper (e.g., benchmarking, new method, or
biological insight), and analyze them on an individual level, but can also
look at the global set of workflows for a large set of papers (Fig. 1). Thus,
FlaMBé has important downstream potential for extracting knowledge at multiple
levels.

Figure 2: Sankey visualizations of (A) tool and (B) context workflows from an
example paper [41]. Visualizations here focus on one entity at a time, either
the computational tools being used throughout the paper (vertical bars in A)
or the context (vertical bars in B) in which they are being used. Figure 3:
Sankey visualizations of the joint tool-context workflow from an example paper
[42]. Visualization here depicts the workflow of (tool, context) pairs
(vertical bars), where context is denoted within the parentheses.

###  4.1 Potential downstream applications

There are many other interesting downstream applications that FlaMBé can be
used to study. In addition to the advances in developing systematic methods
for procedural knowledge extraction, we want to highlight the scientific value
of improved modeling here. Specifically, structured representations would
potentially allow for improved computational method recommendation depending
on the goals of a particular study, as well as highlight gaps and areas of
need for new computational method development. Importantly, one of the natural
concerns that has been raised in psychology and more recently in machine
learning is that having ever more complex computational workflows can spawn
multiverses. The multiverse represents the set of parallel universes where
slightly different paths (e.g., methods or analysis steps) are taken towards
the same goal. Multiverse analyses are undertaken to see how reliable results
and conclusions are in light of these implicit decisions [43, 44]. We believe
one of the most exciting downstream applications of FlaMBé is systematic
multiverse analyses of the complex workflows undertaken in biomedical
research, towards the ultimate goal of improving transparency and
reproducibility of research claims.

##  5 Limitations and Future Work

One of the current limitations of FlaMBé is that though the number of entity-
level annotations is high, there are relatively fewer examples of the more
complex annotation types of tool context and workflow. We plan to address this
through larger annotation efforts that will further expand these categories.
Because FlaMBé has also proposed a systematic, structured representation that
can be used as input to existing language models, these future efforts can be
aided by computational predictions that can guide manual curation efforts. In
these follow-up efforts, we foresee that the NER-related annotations will be
easiest to automate, followed be NED, with the tool context and workflow
predictions being more challenging. Any automated annotations will be reviewed
by expert curators before release of an updated dataset. We do not foresee
negative societal impacts, though incorrect workflows could potentially be
misleading for downstream research, and thus we would encourage thorough
evaluation of all predictions.

With FlaMBé, we have broken down the more complex, abstract procedural
knowledge extraction problem into more structured declarative knowledge tasks
that the community is already well-equipped to tackle. Intriguingly, cognitive
psychology research has pointed towards the fact that in humans, procedural
and declarative knowledge are intertwined, but can sometimes be learned
independently of one another [45]. Thus, there may also be benefit to using
different, more “procedural” representations for learning. In some sense, one
ML area that has tried to learn and mimic human procedural knowledge is
reinforcement learning. A good example of this is with “script knowledge” [46]
and generally text-based games [47], which have used a game approach to
improve modeling at the intersection of language understanding and complex
decision-making. Reinforcement learning has also found some early success in
reasoning over large scale knowledge graphs. Procedural knowledge extraction
from academic texts could potentially also benefit from this type of
framework. One of the unique aspects of FlaMBé is that though we have
developed a structured representation, they can also tie together (e.g., we
have annotated individual edges that can be viewed jointly as a graph). The
disambiguated terms also tie in with existing knowledge bases that can be
incorporated into knowledge graph research. It will be interesting to see
whether new methods can be developed that could take advantage of the joint
representation and learn more than the sum of the parts.

##  6 Conclusion

In conclusion, we have developed FlaMBé, a collection of datasets that
together form structured representations of procedural knowledge captured in
scientific literature. The dataset provides annotations for 1,195 paper
abstracts and 55 full text papers, spanning nearly 700,000 tokens. In addition
to providing the largest NER and NED dataset for tissue and cell type, we also
provide annotations for computational tool and method, as well as the analysis
task a tool is used in. Finally, we also annotate computational workflows
within papers that can potentially be used in many downstream applications.
Our dataset and associated code are accessible at
https://github.com/ylaboratory/flambe.

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# Closed Form Equations for Triangular Numbers Multiple of Other Triangular
Numbers

Vladimir PLETSER  European Space Agency (ret.)

###### Abstract.

Triangular numbers that are multiple of other triangular numbers are
investigated. It is known that for any positive non-square integer multiplier,
there is an infinity of multiples of triangular numbers which are triangular
numbers. If the multiplier is a squared integer, there is either one or no
solution, depending on the multiplier value. Instead of recurrent relations,
we develop in this paper closed form equations to calculate directly the
values of triangular numbers and their indices without the need of knowing the
previous solutions. We develop the theoretical equations for four cases of
ranks from 1 to 4 and we give several examples for non-square multipliers 2,
3, 5 and 8.

Keywords: Triangular Numbers, Multiple of Triangular Numbers, Closed Form
Equations

##  1\. Introduction

Triangular numbers are defined as $T_{t}=\frac{t\left(t+1\right)}{2}$ and
enjoy many properties, relations and formulas (see e.g. [1, 2]). Triangular
numbers $T_{\xi}$ that are multiples of other triangular number $T_{t}$

(1) $T_{\xi}=kT_{t}$

have been investigated in the past in some specific cases ([3, 4, 5, 6, 7,
8]). Recently, Pletser showed ([9]) that, first, for square integer values of
$k$, there are either no or only one solution of (1), depending on the value
of $k$; and second, for non-square integer values of $k$, there are infinitely
many solutions of (1) and recurrent relations can be found for the four
variables $t,\xi,Tt$ and $T_{\xi}$

(2) $\displaystyle t_{n}$
$\displaystyle=2\left(\kappa+1\right)t_{n-r}-t_{n-2r}+\kappa$ (3)
$\displaystyle\xi_{n}$
$\displaystyle=2\left(\kappa+1\right)\xi_{n-r}-\xi_{n-2r}+\kappa$ (4)
$\displaystyle T_{t_{n}}$
$\displaystyle=\left(4\left(\kappa+1\right)^{2}-2\right)T_{t_{n-r}}-T_{t_{n-2r}}+\left(T_{\kappa}-\gamma\right)$
(5) $\displaystyle T_{\xi_{n}}$
$\displaystyle=\left(4\left(\kappa+1\right)^{2}-2\right)T_{\xi_{n-r}}-T_{\xi_{n-2r}}+k\left(T_{\kappa}-\gamma\right)$

These recurrent relations involve three parameters specific to each case of
the multiplier $k$, i.e., a rank $r$, defined as the number of successive
values of $t$ solutions of (1) with slowly decreasing ratios of the
$r^{\text{th}}$ to the $\left(r-1\right)^{\text{th}}$ values of $t$,
$t_{r-1}/t_{r}$. The two other parameters are $\kappa$ and $\gamma$
respectively the sum $\kappa=t_{r-1}+t_{r}$ and the product
$\gamma=t_{r-1}t_{r}$ of the first two sequential values of $t_{n}$ for
$n=r-1$ and $r$. Several relations exist between these parameters and are
investigated in [9]. Note that only cases with $k>1$ are of interest as $k=0$
and $k=1$ yield obvious solutions respectively $\xi=0$ and $\xi=t$, both
$\forall t$. Although triangular numbers $T_{t}$ are usually defined for
$t\in\mathbb{Z}^{+}$, triangular numbers can be extended to negative indices
$t<0$ as $T_{-t}=T_{t-1}$.

In this paper, instead of solutions with recurrent relations, we investigate
closed form solutions of (1) for the case of non-square integer values of $k$.

##  2\. Closed Form Equations

###  2.1. General equations

One wishes to calculate directly the $n^{\text{th}}$value of
$t_{n},\xi_{n},T_{t_{n}}$ and $T_{\xi_{n}}$ without necessarily knowing
smaller values needed for recurrent solutions. Closed form equations are easy
to calculate from the recursive relations (2) to (5), which are all linear and
non-homogeneous. Let us consider first (2). The associated closed form
equation is the sum of the homogeneous solution $t_{h_{n}}$ and a particular
solution $t_{p}$ (for more details, see e.g. [10, 11, 12]).

The homogeneous characteristic equation associated to (2) reads successively

(6) $\displaystyle x^{2r}-2\left(\kappa+1\right)x^{r}+1$ $\displaystyle=0$ (7)
$\displaystyle\left(x^{r}-\alpha^{r}\right)\left(x^{r}-\beta^{r}\right)$
$\displaystyle=0$

with

(8) $\displaystyle\alpha$
$\displaystyle=\left(\left(\kappa+1\right)+\sqrt{\kappa\left(\kappa+2\right)}\right)^{\frac{1}{r}}$
(9) $\displaystyle\beta$
$\displaystyle=\left(\left(\kappa+1\right)-\sqrt{\kappa\left(\kappa+2\right)}\right)^{\frac{1}{r}}$

with the obvious relation $\left(\alpha\beta\right)^{r}=1$. This equation (7)
has $2r$ distinct characteristic roots, including real and complex roots and,
for odd $r$, $r=2\rho+1$, one has

$\displaystyle x^{r}-\alpha^{r}$
$\displaystyle=\left(x-\alpha\right)\left(\sum_{j=0}^{2\rho}\alpha^{j}x^{2\rho-j}\right)$
(10)
$\displaystyle=\left(x-\alpha\right)\prod_{j=1}^{\rho}\left[x-\alpha\left(\cos\left(\frac{2j\pi}{r}\right)\pm
i\sin\left(\frac{2j\pi}{r}\right)\right)\right]$

and for even $r$, $r=2\rho$,

$\displaystyle x^{r}-\alpha^{r}$
$\displaystyle=\left(x-\alpha\right)\left(x+\alpha\right)\left(\sum_{j=0}^{\rho-1}\alpha^{2j}x^{2\left(\rho-j-1\right)}\right)$
(11)
$\displaystyle=\left(x-\alpha\right)\left(x+\alpha\right)\prod_{j=1}^{\rho-1}\left[x-\alpha\left(\cos\left(\frac{j\pi}{\rho}\right)\pm
i\sin\left(\frac{j\pi}{\rho}\right)\right)\right]$

and similar relations for $\beta$ replacing $\alpha$. Recall that complex
characteristic roots introduce the element of periodicity. The homogeneous
solution is therefore respectively (12) for odd $r$, $r=2\rho+1$ and (13) for
even $r$, $r=2\rho$,

(12) $\displaystyle t_{h_{n}}$
$\displaystyle=\left(A+\sum_{j=1}^{\rho}\left[A_{j}^{\prime}\cos\left(\frac{2nj\pi}{r}\right)+A_{j}^{\prime\prime}\sin\left(\frac{2nj\pi}{r}\right)\right]\right)\alpha^{n}$
$\displaystyle+\left(B+\sum_{j=1}^{\rho}\left[B_{j}^{\prime}\cos\left(\frac{2nj\pi}{r}\right)+B_{j}^{\prime\prime}\sin\left(\frac{2nj\pi}{r}\right)\right]\right)\beta^{n}$
(13) $\displaystyle t_{h_{n}}$
$\displaystyle=\left(A+\left(-1\right)^{n}A^{\prime}+\sum_{j=1}^{\rho-1}\left[A_{j}^{\prime\prime}\cos\left(\frac{nj\pi}{\rho}\right)+A_{j}^{\prime\prime\prime}\sin\left(\frac{nj\pi}{\rho}\right)\right]\right)\alpha^{n}$
$\displaystyle+\left(B+\left(-1\right)^{n}B^{\prime}+\sum_{j=1}^{\rho}\left[B_{j}^{\prime\prime}\cos\left(\frac{nj\pi}{\rho}\right)+B_{j}^{\prime\prime\prime}\sin\left(\frac{nj\pi}{\rho}\right)\right]\right)\beta^{n}$

with the $2r$ constants
$A,A^{\prime},A_{j}^{\prime\prime},A_{j}^{\prime\prime\prime},B,B^{\prime},B_{j}^{\prime\prime}$
and $B_{j}^{\prime\prime\prime}$ to be determined by $2r$ boundary conditions,
namely the first $2r$ values of $t_{n}$, for $n=0$ to $n=2r-1$. In practice,
one can simplify this method by using the fact that
$t_{j}=t_{-\left(j+1\right)}$, i.e. for $n=-r$ to $n=r-1$, necessitating to
only know the first $\left(r-1\right)$ values of $t_{n}$.

The particular solution $t_{p}$ can be found by posing it equal to a constant
$t_{p}=\tau$. This particular solution must be a solution of the initial
equation (2). Therefore replacing $t_{n},t_{n-r}$ and $t_{n-2r}$ by
$t_{p}=\tau$ in (2) yields directly $\tau=-1/2$. The complete solution is then
$t_{n}=t_{h_{n}}-1/2$.

###  2.2. Closed form equations for $r=1$ ($\rho=0$)

The homogeneous solution (12) reduces to

(14) $t_{h_{n}}=A\alpha^{n}+B\beta^{n}$

yielding the complete solution

(15) $t_{n}=A\alpha^{n}+B\beta^{n}-1/2$

with the boundary conditions $t_{0}=t_{-1}=0$, yielding $A+B=1/2$ and
$A\alpha^{-1}+B\beta^{-1}=1/2$, giving
$A=\alpha\left(1-\beta\right)/2\left(\alpha-\beta\right)$ and
$B=\beta\left(\alpha-1\right)/2\left(\alpha-\beta\right)$.

####  2.2.1. Closed forms for $k=2$

The smallest case of $k$ having a rank equal to unity is $k=2$, for which
$t_{r-1}=0$, $t_{r}=2$ (see OEIS [13], A053141) and $\kappa=2$, yielding from
(8) and (9), $\alpha=\left(3+2\sqrt{2}\right)=\left(1+\sqrt{2}\right)^{2}$,
$\beta=\left(3-2\sqrt{2}\right)=\left(1-\sqrt{2}\right)^{2}$, and further,
$A=\left(2+\sqrt{2}\right)/8$, $B=\left(2-\sqrt{2}\right)/8$, and the closed
form equation

(16)
$t_{n}=\frac{1}{8}\left(\left(2+\sqrt{2}\right)\left(1+\sqrt{2}\right)^{2n}+\left(2-\sqrt{2}\right)\left(1-\sqrt{2}\right)^{2n}\right)-\frac{1}{2}$

The closed form equations for the three other variables for this case of $k=2$
are found similarly:

\- for $\xi_{n}$ (3) (see OEIS [13], A001652)

(17)
$\xi_{n}=\frac{1}{4}\left(\left(1+\sqrt{2}\right)^{2n+1}+\left(1-\sqrt{2}\right)^{2n+1}\right)-\frac{1}{2}$

\- for $T_{t_{n}}$ (4) (see OEIS [13], A075528), with $\left(17\pm
12\sqrt{2}\right)=\left(1\pm\sqrt{2}\right)^{4}$,

(18)
$T_{t_{n}}=\frac{1}{64}\left(\left(1+\sqrt{2}\right)^{4n+2}+\left(1-\sqrt{2}\right)^{4n+2}\right)-\frac{3}{32}$

\- for $T_{\xi_{n}}$ (5) (see OEIS [13], A029549)

(19)
$T_{\xi_{n}}=\frac{1}{32}\left(\left(1+\sqrt{2}\right)^{4n+2}+\left(1-\sqrt{2}\right)^{4n+2}\right)-\frac{3}{16}$

####  2.2.2. Closed forms for $k=3$

For $k=3$, $r=1$ and $t_{r-1}=0$, $t_{r}=1$ (see OEIS [13], A061278) and
$\kappa=1$, yielding $\alpha=\left(2+\sqrt{3}\right)$,
$\beta=\left(2-\sqrt{3}\right)$ from (8) and (9), and
$A=\left(3+\sqrt{3}\right)/12$, $B=\left(3-\sqrt{3}\right)/12$, and the closed
form

(20)
$t_{n}=\frac{1}{12}\left(\left(3+\sqrt{3}\right)\left(2+\sqrt{3}\right)^{n}+\left(3-\sqrt{3}\right)\left(2-\sqrt{3}\right)^{n}\right)-\frac{1}{2}$

Similarly, the closed forms for the three other variables are:

\- for $\xi_{n}$ (3) (see OEIS [13], A001571):

(21)
$\xi_{n}=\frac{1}{4}\left(\left(1+\sqrt{3}\right)\left(2+\sqrt{3}\right)^{n}+\left(1-\sqrt{3}\right)\left(2-\sqrt{3}\right)^{n}\right)-\frac{1}{2}$

\- for $T_{t_{n}}$ (4) (see OEIS [13], A076139), with $\left(7\pm
4\sqrt{3}\right)=\left(2\pm\sqrt{3}\right)^{2}$:

(22)
$T_{t_{n}}=\frac{1}{48}\left(\left(2+\sqrt{3}\right)^{2n+1}+\left(2-\sqrt{3}\right)^{2n+1}\right)-\frac{1}{12}$

\- for $T_{\xi_{n}}$ (5) (see OEIS [13], A076140):

(23)
$T_{\xi_{n}}=\frac{1}{16}\left(\left(2+\sqrt{3}\right)^{2n+1}+\left(2-\sqrt{3}\right)^{2n+1}\right)-\frac{1}{4}$

###  2.3. Closed form equations for $r=2$ ($\rho=1$)

The homogeneous solution (13) reduces to

(24)
$t_{h_{n}}=\left(A+\left(-1\right)^{n}A^{\prime}\right)\alpha^{n}+\left(B+\left(-1\right)^{n}B^{\prime}\right)\beta^{n}$

yielding the complete solution

(25)
$t_{n}=\left(A+\left(-1\right)^{n}A^{\prime}\right)\alpha^{n}+\left(B+\left(-1\right)^{n}B^{\prime}\right)\beta^{n}-1/2$

with the boundary conditions $t_{0}=t_{-1}=0$ and $t_{1}=t_{-2}$, yielding the
four relations

$\displaystyle A+A^{\prime}+B+B^{\prime}$ $\displaystyle=$ $\displaystyle 1/2$
$\displaystyle\left(A-A^{\prime}\right)\alpha^{-1}+\left(B-B^{\prime}\right)\beta^{-1}$
$\displaystyle=$ $\displaystyle 1/2$
$\displaystyle\left(A-A^{\prime}\right)\alpha+\left(B-B^{\prime}\right)\beta$
$\displaystyle=$ $\displaystyle t_{1}+1/2$
$\displaystyle\left(A+A^{\prime}\right)\alpha^{-2}+\left(B+B^{\prime}\right)\beta^{-2}$
$\displaystyle=$ $\displaystyle t_{1}+1/2$

yielding

$\displaystyle A$
$\displaystyle=\frac{\alpha\left(\alpha+1\right)\left(1-\beta^{2}\right)+2t_{1}\left(\alpha-1\right)}{4\left(\alpha^{2}-\beta^{2}\right)}$
$\displaystyle A^{\prime}$
$\displaystyle=\frac{\alpha\left(\alpha-1\right)\left(1-\beta^{2}\right)-2t_{1}\left(\alpha+1\right)}{4\left(\alpha^{2}-\beta^{2}\right)}$
$\displaystyle B$
$\displaystyle=\frac{\beta\left(\beta+1\right)\left(\alpha^{2}-1\right)-2t_{1}\left(\beta-1\right)}{4\left(\alpha^{2}-\beta^{2}\right)}$
$\displaystyle B^{\prime}$
$\displaystyle=\frac{\beta\left(\beta-1\right)\left(\alpha^{2}-1\right)+2t_{1}\left(\beta+1\right)}{4\left(\alpha^{2}-\beta^{2}\right)}$

####  2.3.1. Closed forms for $k=5$

The smallest case of $k$ with rank two is $k=5$ (see OEIS [13], A077259), for
which $\kappa=8$ yielding $\alpha=\sqrt{9+4\sqrt{5}}=2+\sqrt{5}$ and
$\beta=\sqrt{9-4\sqrt{5}}=2-\sqrt{5}$, with $t_{1}=2$. The complete solution
is then

(26) $\displaystyle t_{n}$
$\displaystyle=\frac{1}{20}\left(\left(\left(5+2\sqrt{5}\right)-\left(-1\right)^{n}\sqrt{5}\right)\left(2+\sqrt{5}\right)^{n}\right.$
$\displaystyle\left.+\left(\left(5-2\sqrt{5}\right)+\left(-1\right)^{n}\sqrt{5}\right)\left(2-\sqrt{5}\right)^{n}\right)-\frac{1}{2}$

which yields for even $n$

(27)
$t_{n}=\frac{1}{20}\left(\left(5+\sqrt{5}\right)\left(2+\sqrt{5}\right)^{n}+\left(5-\sqrt{5}\right)\left(2-\sqrt{5}\right)^{n}\right)-\frac{1}{2}$

and for odd $n$

(28)
$t_{n}=\frac{1}{20}\left(\left(5+3\sqrt{5}\right)\left(2+\sqrt{5}\right)^{n}+\left(5-3\sqrt{5}\right)\left(2-\sqrt{5}\right)^{n}\right)-\frac{1}{2}$

####  2.3.2. Closed forms for $k=8$

For $k=8$ (see OEIS [13], A336623), $r=2$, for which $\kappa=16$ yielding
$\alpha=\sqrt{17+12\sqrt{2}}=\left(1+\sqrt{2}\right)^{2}$ and
$\beta=\sqrt{17-12\sqrt{2}}=\left(1-\sqrt{2}\right)^{2}$, with $t_{1}=5$. The
complete solution is then

(29) $\displaystyle t_{n}$
$\displaystyle=\frac{1}{8}\left(\left(\frac{3}{2}\left(2+\sqrt{2}\right)-\left(-1\right)^{n}\left(1+\sqrt{2}\right)\right)\left(1+\sqrt{2}\right)^{2n}\right.$
$\displaystyle\left.+\left(\frac{3}{2}\left(2-\sqrt{2}\right)-\left(-1\right)^{n}\left(1-\sqrt{2}\right)\right)\left(1-\sqrt{2}\right)^{2n}\right)-\frac{1}{2}$

which can be dissociated for even $n$, yielding

(30)
$t_{n}=\frac{1}{16}\left(\left(4+\sqrt{2}\right)\left(1+\sqrt{2}\right)^{2n}+\left(4-\sqrt{2}\right)\left(1-\sqrt{2}\right)^{2n}\right)-\frac{1}{2}$

and for odd $n$, yielding

(31)
$t_{n}=\frac{1}{16}\left(\left(8+5\sqrt{2}\right)\left(1+\sqrt{2}\right)^{2n}+\left(8-5\sqrt{2}\right)\left(1-\sqrt{2}\right)^{2n}\right)-\frac{1}{2}$

Similarly, the closed forms for the three other variables are:

\- for $\xi_{n}$ (3) (see OEIS [13], A336625):

(32)
$\xi_{n}=\frac{1}{4}\left(\left(3-(-1)^{n}\sqrt{2}\right)\left(1+\sqrt{2}\right)^{2n+1}+\left(3+(-1)^{n}\sqrt{2}\right)\left(1-\sqrt{2}\right)^{2n+1}\right)-\frac{1}{2}$

yielding for even $n$ (upper sign) and odd $n$ (lower sign):

(33)
$\xi_{n}=\frac{1}{4}\left(\left(3\mp\sqrt{2}\right)\left(1+\sqrt{2}\right)^{2n+1}+\left(3\pm\sqrt{2}\right)\left(1-\sqrt{2}\right)^{2n+1}\right)-\frac{1}{2}$

\- for $T_{t_{n}}$ (4) (see OEIS [13], A336624):

(34) $\displaystyle T_{t_{n}}$
$\displaystyle=\frac{1}{256}\left(\left(11-\left(-1\right)^{n}6\sqrt{2}\right)\left(1+\sqrt{2}\right)^{2\left(2n+1\right)}\right.$
$\displaystyle\left.+\left(11+\left(-1\right)^{n}6\sqrt{2}\right)\left(1-\sqrt{2}\right)^{2\left(2n+1\right)}\right)-\frac{9}{128}$

yielding for even $n$ (upper sign) and odd $n$ (lower sign):

(35)
$T_{t_{n}}=\frac{1}{256}\left(\left(3\mp\sqrt{2}\right)^{2}\left(1+\sqrt{2}\right)^{2\left(2n+1\right)}+\left(3\pm\sqrt{2}\right)^{2}\left(1-\sqrt{2}\right)^{2\left(2n+1\right)}\right)-\frac{9}{128}$

\- for $T_{\xi_{n}}$ (5) (see OEIS [13], A336626):

(36) $\displaystyle T_{\xi_{n}}$
$\displaystyle=\frac{1}{32}\left(\left(11\left(1+\sqrt{2}\right)^{2}-(-1)^{n}6\left(4+3\sqrt{2}\right)\right)\left(1+\sqrt{2}\right)^{4n}\right.$
$\displaystyle\left.+\left(11\left(1-\sqrt{2}\right)^{2}-(-1)^{n}6\left(4-3\sqrt{2}\right)\right)\left(1-\sqrt{2}\right)^{4n}\right)-\frac{9}{16}$

yielding for even $n$:

(37)
$T_{\xi_{n}}=\frac{1}{32}\left(\left(1+2\sqrt{2}\right)^{2}\left(1+\sqrt{2}\right)^{4n}+\left(1-2\sqrt{2}\right)^{2}\left(1-\sqrt{2}\right)^{4n}\right)-\frac{9}{16}$

and for odd $n$:

(38)
$T_{\xi_{n}}=\frac{1}{32}\left(\left(5+4\sqrt{2}\right)\left(1+\sqrt{2}\right)^{4n}+\left(5-4\sqrt{2}\right)\left(1-\sqrt{2}\right)^{4n}\right)-\frac{9}{16}$

###  2.4. Closed form equations for $r=3$ ($\rho=1$)

The homogeneous solution (12) yields a complete solution that reduces to

(39) $\displaystyle t_{n}$
$\displaystyle=\left(A+A^{\prime}\cos\left(\frac{2n\pi}{3}\right)+A^{\prime\prime}\sin\left(\frac{2n\pi}{3}\right)\right)\alpha^{n}$
$\displaystyle+\left(B+B^{\prime}\cos\left(\frac{2n\pi}{3}\right)+B^{\prime\prime}\sin\left(\frac{2n\pi}{3}\right)\right)\beta^{n}-1/2$
$\displaystyle=\left(A\alpha^{n}+B\beta^{n}\right)+\left(A^{\prime}\alpha^{n}+B^{\prime}\beta^{n}\right)\cos\left(\frac{2n\pi}{3}\right)$
$\displaystyle+\left(A^{\prime\prime}\alpha^{n}+B^{\prime\prime}\beta^{n}\right)\sin\left(\frac{2n\pi}{3}\right)-1/2$

with the boundary conditions $t_{0}=t_{-1}=0$, $t_{1}=t_{-2}$ and
$t_{2}=t_{-3}$, giving six relations from which one finds the following
expressions of the six constants

$\displaystyle A$
$\displaystyle=\frac{\alpha\left(\alpha^{2}-\beta^{3}\right)+\left(2t_{1}+1\right)\alpha^{2}\left(1-\beta^{3}\right)+\left(2t_{2}+1\right)\left(\alpha-1\right)}{6\left(\alpha^{3}-\beta^{3}\right)}$
$\displaystyle A^{\prime}$
$\displaystyle=\frac{\alpha\left(2\alpha+\beta^{3}\right)-\left(2t_{1}+1\right)\alpha^{2}\left(1-\beta^{3}\right)-\left(2t_{2}+1\right)\left(\alpha+2\right)}{6\left(\alpha^{3}-\beta^{3}\right)}$
$\displaystyle A^{\prime\prime}$
$\displaystyle=\frac{\sqrt{3}\alpha\left(\beta^{3}+\left(2t_{1}+1\right)\alpha\left(1-\beta^{3}\right)-\left(2t_{2}+1\right)\right)}{6\left(\alpha^{3}-\beta^{3}\right)}$
$\displaystyle B$
$\displaystyle=\frac{\beta\left(\alpha^{3}-\beta^{2}\right)+\left(2t_{1}+1\right)\beta^{2}\left(\alpha^{3}-1\right)-\left(2t_{2}+1\right)\left(\beta-1\right)}{6\left(\alpha^{3}-\beta^{3}\right)}$
$\displaystyle B^{\prime}$
$\displaystyle=\frac{-\beta\left(\alpha^{3}+2\beta^{2}\right)-\left(2t_{1}+1\right)\beta^{2}\left(\alpha^{3}-1\right)+\left(2t_{2}+1\right)\left(\beta+2\right)}{6\left(\alpha^{3}-\beta^{3}\right)}$
$\displaystyle B^{\prime\prime}$
$\displaystyle=\frac{\sqrt{3}\beta\left(-\alpha^{3}+\left(2t_{1}+1\right)\beta\left(\alpha^{3}-1\right)+\left(2t_{2}+1\right)\right)}{6\left(\alpha^{3}-\beta^{3}\right)}$

The smallest case of $k$ with rank three is $k=10$, for which $\kappa=18$
yielding
$\alpha=\left(19+6\sqrt{10}\right)^{\frac{1}{3}}=\left(3+\sqrt{10}\right)^{\frac{2}{3}}$
and
$\beta=\left(19-6\sqrt{10}\right)^{\frac{1}{3}}=\left(3-\sqrt{10}\right)^{\frac{2}{3}}$,
with $t_{1}=1$, $t_{2}=6$.

###  2.5. Closed form equations for $r=4$ ($\rho=2$)

The homogeneous solution (13) yields a complete solution that reduces to

(40) $\displaystyle t_{n}$
$\displaystyle=\left(A+\left(-1\right)^{n}A^{\prime}+A^{\prime\prime}\cos\left(\frac{n\pi}{2}\right)+A^{\prime\prime\prime}\sin\left(\frac{n\pi}{2}\right)\right)\alpha^{n}$
$\displaystyle+\left(B+\left(-1\right)^{n}B^{\prime}+B^{\prime\prime}\cos\left(\frac{n\pi}{2}\right)+B^{\prime\prime\prime}\sin\left(\frac{n\pi}{2}\right)\right)\beta^{n}-1/2$

and that simplifies as follows:

\- for $n\equiv 0\,\text{mod$4$}$,
$t_{n}=\left(A+A^{\prime}+A^{\prime\prime}\right)\alpha^{n}+\left(B+B^{\prime}+B^{\prime\prime}\right)\beta^{n}-1/2$

\- for $n\equiv 1\,\text{mod$4$}$,
$t_{n}=\left(A-A^{\prime}+A^{\prime\prime\prime}\right)\alpha^{n}+\left(B-B^{\prime}+B^{\prime\prime\prime}\right)\beta^{n}-1/2$

\- for $n\equiv 2\,\text{mod$4$}$,
$t_{n}=\left(A+A^{\prime}-A^{\prime\prime}\right)\alpha^{n}+\left(B+B^{\prime}-B^{\prime\prime}\right)\beta^{n}-1/2$

\- for $n\equiv 3\,\text{mod$4$}$,
$t_{n}=\left(A-A^{\prime}-A^{\prime\prime\prime}\right)\alpha^{n}+\left(B-B^{\prime}-B^{\prime\prime\prime}\right)\beta^{n}-1/2$

with the boundary conditions $t_{0}=t_{-1}=0$, $t_{1}=t_{-2}$, $t_{2}=t_{-3}$
and $t_{3}=t_{-4}$, giving eight relations yielding the eight constants

$\displaystyle A$
$\displaystyle=\frac{\alpha\left(\alpha^{3}-\beta^{4}\right)+\left(2t_{1}+1\right)\alpha^{2}\left(\alpha-\beta^{4}\right)+\left(2t_{2}+1\right)\alpha^{2}\left(1-\alpha\beta^{4}\right)+\left(2t_{3}+1\right)\left(\alpha-1\right)}{8\left(\alpha^{4}-\beta^{4}\right)}$
$\displaystyle A^{\prime}$
$\displaystyle=\frac{\alpha\left(\alpha^{3}+\beta^{4}\right)-\left(2t_{1}+1\right)\alpha^{2}\left(\alpha+\beta^{4}\right)+\left(2t_{2}+1\right)\alpha^{2}\left(1+\alpha\beta^{4}\right)-\left(2t_{3}+1\right)\left(\alpha+1\right)}{8\left(\alpha^{4}-\beta^{4}\right)}$
$\displaystyle A^{\prime\prime}$
$\displaystyle=\frac{\alpha^{4}+\left(2t_{1}+1\right)\alpha^{2}\beta^{4}-\left(2t_{2}+1\right)\alpha^{2}-\left(2t_{3}+1\right)}{4\left(\alpha^{4}-\beta^{4}\right)}$
$\displaystyle A^{\prime\prime\prime}$
$\displaystyle=\frac{\alpha\left(\beta^{4}+\left(2t_{1}+1\right)\alpha^{2}-\left(2t_{2}+1\right)\alpha^{2}\beta^{4}-\left(2t_{3}+1\right)\right)}{4\left(\alpha^{4}-\beta^{4}\right)}$
$\displaystyle B$
$\displaystyle=\frac{\beta\left(\alpha^{4}-\beta^{3}\right)+\left(2t_{1}+1\right)\beta^{2}\left(\alpha^{4}-\beta\right)+\left(2t_{2}+1\right)\beta^{2}\left(\alpha^{4}\beta-1\right)-\left(2t_{3}+1\right)\left(\beta-1\right)}{8\left(\alpha^{4}-\beta^{4}\right)}$
$\displaystyle B^{\prime}$
$\displaystyle=\frac{-\beta\left(\alpha^{4}+\beta^{3}\right)+\left(2t_{1}+1\right)\beta^{2}\left(\alpha^{4}+\beta\right)-\left(2t_{2}+1\right)\beta^{2}\left(\alpha^{4}\beta+1\right)+\left(2t_{3}+1\right)\left(\beta+1\right)}{8\left(\alpha^{4}-\beta^{4}\right)}$
$\displaystyle B^{\prime\prime}$
$\displaystyle=\frac{-\beta^{4}-\left(2t_{1}+1\right)\alpha^{4}\beta^{2}+\left(2t_{2}+1\right)\beta^{2}+\left(2t_{3}+1\right)}{4\left(\alpha^{4}-\beta^{4}\right)}$
$\displaystyle B^{\prime\prime\prime}$
$\displaystyle=\frac{\beta\left(-\alpha^{4}-\left(2t_{1}+1\right)\beta^{2}+\left(2t_{2}+1\right)\alpha^{4}\beta^{2}+\left(2t_{3}+1\right)\right)}{4\left(\alpha^{4}-\beta^{4}\right)}$

The smallest case of $k$ with rank four is $k=13$, for which $\kappa=648$
yielding
$\alpha=\left(649+180\sqrt{13}\right)^{\frac{1}{4}}=\left(18+5\sqrt{13}\right)^{\frac{1}{2}}$
and
$\beta=\left(649-180\sqrt{13}\right)^{\frac{1}{4}}=\left(18-5\sqrt{13}\right)^{\frac{1}{2}}$,
with $t_{1}=3$, $t_{2}=21$ and $t_{3}=234$.

##  3\. Conclusions

We have shown that for triangular numbers multiple of other triangular
numbers, closed form equations can be developed instead of recurrent
relations, to calculate directly the values of triangular numbers and their
indices without knowing the whole preceding sequence of values. We developed
the theoretical equations for four cases of ranks from 1 to 4 and we applied
them in several examples for the non-square multipliers 2, 3, 5 and 8.

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