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 A380558 #12 Feb 15 2025 11:16:44
%S A380558 1,2,10,62,469,4028,37984,385202,4144798,46882400,553733875,
%T A380558 6795347708,86314711993,1131422763410,15268625617174,211726229534738,
%U A380558 3012057754693912,43903115899714844,654923002676505376,9989373316478767304,155663132037403882606,2476418549848925209424,40195761790035415573939
%N A380558 G.f. A(x) satisfies A(x - A(x)) = x^2/(1 - x^2).
%C A380558 Conjecture: a(n) is odd iff n = 2*A004760(k) for some k > 1, where A004760 lists numbers whose binary expansion does not begin 10.
%H A380558 Paul D. Hanna, <a href="/A380558/b380558.txt">Table of n, a(n) for n = 2..520</a>
%F A380558 G.f. A(x) = Sum_{n>=2} a(n)*x^n satisfies the following formulas.
%F A380558 (1) A(x - A(x)) = x^2/(1 - x^2).
%F A380558 (2) A(x) = B(x)^2/(1 - B(x)^2) where B(x) = x + A(B(x)) and B(x - A(x)) = x.
%F A380558 (3) A(x) = B(x)^2/(1 - B(x)^2) where B(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n / n!.
%e A380558 G.f.: A(x) = x^2 + 2*x^3 + 10*x^4 + 62*x^5 + 469*x^6 + 4028*x^7 + 37984*x^8 + 385202*x^9 + 4144798*x^10 + 46882400*x^11 + 553733875*x^12 + ...
%e A380558 where A(x - A(x)) = x^2/(1 - x^2).
%e A380558 Let B(x) = Series_Reversion(x - A(x)), where
%e A380558 B(x) = x + x^2 + 4*x^3 + 25*x^4 + 190*x^5 + 1645*x^6 + 15652*x^7 + 160186*x^8 + 1739032*x^9 + 19838179*x^10 + ... + A380678(n)*x^n + ...
%e A380558 then B(x) = x + A(B(x)).
%o A380558 (PARI) /* Generates N terms of this sequence */
%o A380558 N = 40; A=x^2; for(m=1,N, A=truncate(A); B = serreverse(x - A +x*O(x^m)); A = B^2/(1-B^2) ); Vec(A)
%Y A380558 Cf. A380678, A276370, A004760.
%K A380558 nonn
%O A380558 2,2
%A A380558 _Paul D. Hanna_, Feb 13 2025