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 A168505 #27 Feb 07 2025 00:51:08
%S A168505 1,1,0,-1,-2,-2,0,5,12,16,6,-32,-102,-170,-130,199,966,1978,2192,-650,
%T A168505 -9292,-23624,-33760,-12138,84440,280852,493932,397668,-639676,
%U A168505 -3248464,-6947460,-8068587,2165980,35591960,94129446,139864828,56393482,-352505722
%N A168505 Expansion of 1/(1-x/(1+x/(1-x/(1-x/(1+x/(1-x/(1-x/(1+x/(1-... (continued fraction).
%C A168505 Hankel transform is A131561(n+1). First column of array whose production matrix begins
%C A168505 1, 1;
%C A168505 -1, 0, 1;
%C A168505 0, 1, 0, 1;
%C A168505 0, 0, -1, 2, 1;
%C A168505 0, 0, 0, -1, 0, 1;
%C A168505 0, 0, 0, 0, 1, 0, 1;
%C A168505 0, 0, 0, 0, 0, -1, 2, 1;
%F A168505 G.f.: 1/(1-x+x^2/(1-x^2/(1+x^2/(1-2x+x^2/(1-x^2/(1+x^2/(1-2x+x^2/(1-x^2/(1+... (continued fraction, defined by the sequences (1,0,0,2,0,0,2,0,0,2,0,...) and (-1,1,-1,-1,1,-1,...));
%F A168505 g.f.: (1+x-sqrt(1-2x+x^2+4x^3))/(2x(1-x)).
%F A168505 a(n) = Sum_{k=0..n} A198379(n,k)*(-1)^(n-k). - _Philippe Deléham_, Oct 29 2011
%F A168505 a(n) = (-1)^n*Sum_{k=0..n} A174014(n,k)*(-2)^k. - _Philippe Deléham_, Feb 16 2012
%F A168505 G.f.: (1+x)/(G(0)+x), where G(k) = 1 - x + x^3/G(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jul 29 2013
%F A168505 Conjecture: (n+1)*a(n) - 3*n*a(n-1) + 3*(n-1)*a(n-2) + 3*(n-4)*a(n-3) + 2*(-2*n+7)*a(n-4) = 0. - _R. J. Mathar_, Feb 10 2015
%F A168505 G.f. A(x) satisfies (A(x) - 1) / A(x)^2 = (x - x^2) / (1 + x). - _Michael Somos_, Jan 20 2017
%F A168505 0 = a(n)*(+16*a(n+1) - 6*a(n+2) - 42*a(n+3) + 54*a(n+4) - 22*a(n+5)) + a(n+1)*(-18*a(n+1) + 27*a(n+2) + 6*a(n+3) - 31*a(n+4) + 18*a(n+5))+ a(n+2)*(-18*a(n+2) + 36*a(n+3) - 30*a(n+4) + 9*a(n+5)) + a(n+3)*(+6*a(n+4) - 6*a(n+5)) + a(n+4)*(+a(n+5)) if n >= 0. - _Michael Somos_, Jan 20 2017
%e A168505 G.f. = 1 + x - x^3 - 2*x^4 - 2*x^5 + 5*x^7 + 12*x^8 + 16*x^9 + 6*x^10 + ...
%o A168505 (PARI) {a(n) = if( n<0, 0, polcoeff( (1 + x - sqrt(1 - 2*x + x^2 + 4*x^3 + x^2 * O(x^n))) / (2*x*(1 - x)), n))}; /* _Michael Somos_, Jan 20 2017 */
%K A168505 easy,sign
%O A168505 0,5
%A A168505 _Paul Barry_, Nov 27 2009