<s>[INST] Pythagorean triplets of the form $a^2+(a+1)^2=c^2$ and the space between them I was searching for pythagorean triples where $b=a+1$, and I found using a python program I made the first 10 integer solutions:


*

*$0^2+1^2=1^2$

*$3^2+4^2=5^2$

*$20^2+21^2=29^2$

*$119^2+120^2=169^2$

*$696^2+697^2=985^2$

*$4059^2+4060^2=5741^2$

*$23660^2+23661^2=33461^2$

*$137903^2+137904^2=195025^2$

*$803760^2+803761^2=1136689^2$

*$4684659^2+4684660^2=6625109^2$


Now what's so interesting? I discovered that any $c$, divided by the previous (for example $5/1$ or $29/5$) limits to $5.828427...=\left(\frac{1}{\sqrt2-1}\right)^2=\sqrt8+3$. My question: why?
 [/INST] The values $\; 1,5,29,169,\cdots\;$ are subset of pell number as shown in the OEIS sequence A000129. You are in fact using the odd-value numbers in the sequence.
If we define Euclid's formula for generating Pythagorean triples as:
$$ \qquad A=m^2-k^2\qquad B=2mk \qquad C=m^2+k^2\qquad$$
the $(m,k)$ values needed to generate these are also Pell numbers and the $n^{th}$ triple can be generated by
\begin{equation}
 m_n= \frac{(1 + \sqrt{2})^{n+1} - (1 - \sqrt{2})^{n+1}}{2\sqrt{2}}\qquad 
 k_n= \frac{(1 + \sqrt{2})^n - (1 - \sqrt{2})^n}{2\sqrt{2}}
 \end{equation}
If you plug these into $\quad C=m^2+k^2\quad $ and take the ratios between adjacent $C$-value polynomials, it should explain the result you have observed, much as ratios of consecutive Fibonacci number converge to $\phi$.
\begin{align*}
\frac{(1 + \sqrt{2})^{2} - (1 - \sqrt{2})^{2}}{2\sqrt{2}}=2 \qquad& 
     \frac{(1 + \sqrt{2})^1 - (1 - \sqrt{2})^1}{2\sqrt{2}}=1 \\ 
&\qquad\qquad  F(2,1)=(3,4,5)\\    
 \frac{(1 + \sqrt{2})^{3} - (1 - \sqrt{2})^{3}}{2\sqrt{2}}=5 \qquad &
     \frac{(1 + \sqrt{2})^2 - (1 - \sqrt{2})^2}{2\sqrt{2}}=2 \\ 
&\qquad\qquad  F(5,2)=(21,20,29)\\
  \frac{(1 + \sqrt{2})^{4} - (1 - \sqrt{2})^{4}}{2\sqrt{2}}=12 \qquad & 
      \frac{(1 + \sqrt{2})^3 - (1 - \sqrt{2})^3}{2\sqrt{2}}=5 \\ 
&\qquad\qquad  F(12,5)=(119,120,169)\\ 
 \frac{(1 + \sqrt{2})^{5} - (1 - \sqrt{2})^{5}}{2\sqrt{2}}=29 \qquad &
     \frac{(1 + \sqrt{2})^4 - (1 - \sqrt{2})^4}{2\sqrt{2}}=12 \\ 
&\qquad\qquad  F(29,12)=(697,696,985)
 \end{align*}
</s>