Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #33 Mar 01 2022 12:00:10
%S 1,-1,1,-1,1,-1,1,-1,2,-3,4,-5,6,-7,8,-10,13,-17,22,-28,35,-43,53,-66,
%T 83,-105,133,-168,211,-264,330,-413,518,-651,819,-1030,1294,-1624,
%U 2037,-2555,3206,-4025,5055,-6349,7973,-10010,12565,-15771,19796,-24851,31200,-39173,49183,-61748,77519,-97315,122166
%N G.f. satisfies: A(x) = 1/(1 + x*A(x^7)) and also the continued fraction: 1 + x*A(x^8) = [1; 1/x, 1/x^7, 1/x^49, 1/x^343, ..., 1/x^(7^(n-1)), ...].
%F This was conjectured to have g.f. (1+x^7) / (1+x+x^7) by _Ralf Stephan_, May 17 2007, but this is wrong. This g.f. produces a sequence which differs at a(57) = -153367. The g.f. gives a(57) = -153366. - _Johannes W. Meijer_, Aug 08 2011
%F a(0) = 1; a(n) = -Sum_{k=0..floor((n-1)/7)} a(k) * a(n-7*k-1). - _Ilya Gutkovskiy_, Mar 01 2022
%p nmax:=57: kmax:=nmax: for k from 0 to kmax do A:= proc(x): add(A101917(n)*x^n, n=0..k) end: f(x):=series(1/(1 + x*A(x^7)),x,k+1); for n from 0 to k do x(n):=coeff(f(x),x,n) od: A101917(k):=x(k): od: seq(A101917(n), n=0..nmax); # _Johannes W. Meijer_, Aug 08 2011
%t m = 57; A[_] = 0; Do[A[x_] = 1/(1 + x A[x^7]) + 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\7+1, A=1/(1+x*subst(A,x,x^7)+x*O(x^n)));polcoeff(A,n,x)
%o (PARI) a(n)=local(M=contfracpnqn(concat(1, vector(ceil(log(n+1)/log(7))+1,n,1/x^(7^(n-1)))))); polcoeff(M[1,1]/M[2,1]+x*O(x^(8*n+1)),8*n+1)
%Y Cf. A101912, A101913, A101914, A101915, A101916, A101918.
%K sign
%O 0,9
%A _Paul D. Hanna_, Dec 20 2004