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%I #23 Mar 01 2022 12:00:25
%S 1,-1,1,-1,1,0,-1,2,-3,3,-2,0,3,-6,8,-8,5,1,-9,17,-22,20,-10,-8,31,
%T -51,60,-50,16,38,-100,150,-163,119,-11,-147,315,-432,433,-268,-70,
%U 522,-964,1222,-1118,542,484,-1756,2887,-3385,2793,-879,-2176,5678,-8472,9186,-6672,542,8372,-17816,24384,-24350,14952
%N G.f. satisfies: A(x) = 1/(1 + x*A(x^4)) and also the continued fraction: 1 + x*A(x^5) = [1; 1/x, 1/x^4, 1/x^16, 1/x^64, ..., 1/x^(4^(n-1)), ...].
%C Sequence appears to have a rational g.f. - _Ralf Stephan_, May 17 2007
%C The conjecture is wrong. The g.f. is dependent on the number of terms. With twenty terms the g.f. is (1 + x^4)/(1 + x + x^4). - _Johannes W. Meijer_, Aug 08 2011
%F a(0) = 1; a(n) = -Sum_{k=0..floor((n-1)/4)} a(k) * a(n-4*k-1). - _Ilya Gutkovskiy_, Mar 01 2022
%p nmax:=62: kmax:=nmax: for k from 0 to kmax do A:= proc(x): add(A101914(n)*x^n, n=0..k) end: f(x):=series(1/(1 + x*A(x^4)),x,k+1); for n from 0 to k do x(n):=coeff(f(x),x,n) od: A101914(k):=x(k): od: seq(A101914(n), n=0..nmax); # _Johannes W. Meijer_, Aug 08 2011
%t m = 63; A[_] = 0; Do[A[x_] = 1/(1 + x A[x^4]) + O[x]^m // Normal, {m}]; CoefficientList[A[x], x] (* _Jean-François Alcover_, Nov 03 2019 *)
%o (PARI) a(n)=local(A);A=1-x;for(i=1,n\4+1, A=1/(1+x*subst(A,x,x^4)+x*O(x^n)));polcoeff(A,n,x)
%o (PARI) a(n)=local(M=contfracpnqn(concat(1, vector(ceil(log(n+1)/log(4))+1,n,1/x^(4^(n-1)))))); polcoeff(M[1,1]/M[2,1]+x*O(x^(5*n+1)),5*n+1)
%Y Cf. A101912, A101913, A101915, A101916, A101917, A101918.
%K sign
%O 0,8
%A _Paul D. Hanna_, Dec 20 2004
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