This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A144229 #33 Mar 19 2018 04:07:00
%S A144229 1,1,4,25,1681,5317636,66314914699609,8947678119828215014722891025,
%T A144229 178098260698995011212395018312912894502905113202338936836
%N A144229 The numerators of the convergents to the recursion x=1/(x^2+1).
%C A144229 The recursion converges to the real root of 1/(x^2+1) - x = 0, 0.682327803...
%C A144229 An interesting consequence of this result occurs if we multiply by x^2+1 to get 1-x-x^3=0. These different equations intersect at the same root 0.682327803... Note also that a(n) is a square. The square roots form sequence A076725.
%C A144229 a(n) is the number of (0,1)-labeled perfect binary trees of height n such that no adjacent nodes have 1 as the label and the root is labeled 0. - _Ran Pan_, May 22 2015
%H A144229 Ran Pan, <a href="http://www.math.ucsd.edu/~projectp/warmups/eR.html">Exercise R</a>, Project P.
%F A144229 a(n+2) = (a(n)^2 + a(n+1))^2. - _Ran Pan_, May 22 2015
%F A144229 a(n) ~ c * d^(2^n), where c = A088559 = 0.465571231876768... is the root of the equation c*(1+c)^2 = 1, d = 1.6634583970724267140029... . - _Vaclav Kotesovec_, May 22 2015
%t A144229 f[n_]:=(n+1/n)/n;Prepend[Denominator[NestList[f,2,7]],1] (* _Vladimir Joseph Stephan Orlovsky_, Nov 19 2010 *)
%t A144229 RecurrenceTable[{a[n]==(a[n-2]^2 + a[n-1])^2, a[0]==1, a[1]==1},a,{n,0,10}] (* _Vaclav Kotesovec_, May 22 2015 after _Ran Pan_ *)
%o A144229 (PARI) x=0;for(j=1,10,x=1/(x^2+1);print1((numerator(x))","))
%Y A144229 Cf. A076725, A088559.
%K A144229 frac,nonn
%O A144229 0,3
%A A144229 _Cino Hilliard_, Sep 15 2008