<s>[INST] Formula for this 2D progression I have two simultaneous progressions and I do not know how to merge them.
the top left entry $f_{1,1}$ is 1.
To move right across the top row : multiply by 64 and add 49
To move down: multiply by 4 and add 1
Resulting in something looking like this:
$$ \left( \begin{array}{ccccc}
1 & 113 & 7281 & \cdots\\
5 & 453 & 29125 &\\
21 & 1813 & 116501 &\\
85 & 7253 & 466005 &\\
\vdots & \vdots & \vdots & \\
\end{array} \right)$$


*

*What I would like is a formula for $f_{a,b}$


typing in these progressions into wolfram I am presented with:
$64n+49 \rightarrow \frac{1}{9}(4^{3a+2}-7)$
$4n+1 \rightarrow \frac{1}{3}(4^{b+1}-1)$


*

*How can I concatenate these formulas to get $f_{a,b}$?


Thank you in advance.
 [/INST] If we have recurrent sequence $$u_1 \quad (\mbox{given}),$$ $$u_{n+1} = a\cdot u_n + b \quad  (\mbox{where } a\ne 1),$$ then
$$
u_{n+1} = au_n+b = a^2u_{n-1}+ab+b = a^3u_{n-2}+a^2b+ab+b \\ = \ldots\\ = a^nu_1+(a^{n-1}+a^{n-2}+\ldots+a^2+a+1)b \\ = a^nu_1 + \dfrac{a^n-1}{a-1}b.
$$
Then top row can be described by the formula
$$
f_{1,b+1} = 1\cdot64^b+\dfrac{64^b-1}{63}\cdot 49 = \dfrac{16\cdot 64^b-7}{9};$$
therefore $f_{a+1,b+1}$ can be described as
$$
f_{a+1,b+1} = \left(\dfrac{16\cdot 64^b-7}{9}\right)\cdot 4^a+\dfrac{4^a-1}{3}\cdot 1 = \dfrac{16\cdot 4^a \cdot 64^b - 4\cdot 4^a - 3}{9} \\ 
= \dfrac{ 4^{a+2} \cdot 64^b - 4^{a+1} - 3}{9} .$$
Example:
$$
f_{4,3} = \dfrac{4^5\cdot 64^2 - 4^4 - 3}{9} = 466005. 
$$
</s>