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A101918 G.f. satisfies: A(x) = 1/(1 + x*A(x^8)) and also the continued fraction: 1 + x*A(x^9) = [1; 1/x, 1/x^8, 1/x^64, 1/x^512, ..., 1/x^(8^(n-1)), ...].

%I A101918 #28 Mar 01 2022 11:59:51
%S A101918 1,-1,1,-1,1,-1,1,-1,1,0,-1,2,-3,4,-5,6,-7,7,-6,4,-1,-3,8,-14,21,-28,
%T A101918 34,-38,39,-36,28,-14,-7,35,-69,107,-146,182,-210,224,-217,182,-113,6,
%U A101918 140,-322,532,-756,973,-1155,1268,-1274,1134,-812,280,476,-1449,2604,-3872,5146,-6280,7092,-7372,6896,-5447,2843,1029
%N A101918 G.f. satisfies: A(x) = 1/(1 + x*A(x^8)) and also the continued fraction: 1 + x*A(x^9) = [1; 1/x, 1/x^8, 1/x^64, 1/x^512, ..., 1/x^(8^(n-1)), ...].
%H A101918 Paul D. Hanna, <a href="/A101918/b101918.txt">Table of n, a(n) for n = 0..1000</a>
%F A101918 The conjecture that this has g.f. (1+x^8) / (1+x+x^8) by _Ralf Stephan_, May 17 2007 is wrong. The first difference occurs at a(73) = -42106. The g.f. gives a(73) = -42105. - _Johannes W. Meijer_, Aug 08 2011
%F A101918 a(0) = 1; a(n) = -Sum_{k=0..floor((n-1)/8)} a(k) * a(n-8*k-1). - _Ilya Gutkovskiy_, Mar 01 2022
%p A101918 nmax:=66: kmax:=nmax: for k from 0 to kmax do A:= proc(x): add(A101918(n)*x^n, n=0..k) end: f(x):=series(1/(1 + x*A(x^8)),x,k+1); for n from 0 to k do x(n):=coeff(f(x),x,n) od: A101918(k):=x(k): od: seq(A101918(n), n=0..nmax); # _Johannes W. Meijer_, Aug 08 2011
%t A101918 nmax = 66; sol = {a[0] -> 1};
%t A101918 Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - (1/(1 + x A[x^8])) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
%t A101918 sol /. Rule -> Set;
%t A101918 a /@ Range[0, nmax] (* _Jean-François Alcover_, Nov 03 2019 *)
%o A101918 (PARI) {a(n)=local(A);A=1-x;for(i=1,n\8+1, A=1/(1+x*subst(A,x,x^8)+x*O(x^n)));polcoeff(A,n,x)}
%o A101918 for(n=0,120,print1(a(n),", "))
%o A101918 (PARI) {a(n)=local(M=contfracpnqn(concat(1, vector(ceil(log(n+1)/log(8))+1,n,1/x^(8^(n-1)))))); polcoeff(M[1,1]/M[2,1]+x*O(x^(9*n+1)),9*n+1)}
%o A101918 for(n=0,120,print1(a(n),", "))
%Y A101918 Cf. A101912-A101917.
%K A101918 sign
%O A101918 0,12
%A A101918 _Paul D. Hanna_, Dec 20 2004