A101916 G.f. satisfies: A(x) = 1/(1 + x*A(x^6)) and also the continued fraction: 1+x*A(x^7) = [1;1/x,1/x^6,1/x^36,1/x^216,...,1/x^(6^(n-1)),...].

1, -1, 1, -1, 1, -1, 1, 0, -1, 2, -3, 4, -5, 5, -4, 2, 1, -5, 10, -15, 19, -21, 20, -15, 5, 10, -29, 50, -70, 85, -90, 80, -51, 1, 69, -154, 244, -324, 375, -376, 307, -153, -91, 414, -788, 1163, -1469, 1621, -1529, 1115, -328, -833, 2299, -3916, 5440, -6550, 6874, -6039, 3741, 170, -5600, 12135, -18990, 25008

Offset: 0

Author

Paul D. Hanna, Dec 20 2004

Keywords

• sign

Comments

This sequence resembles the series expansion of B(x) = (1+x^6)/(1+x+x^6). The first difference occurs at a(43) = 415 versus a(43) = 414. - Johannes W. Meijer, Aug 08 2011

Crossrefs

Cf. A101912-A101915, A101917-A101918.

Programs

Maple

nmax:=63: kmax:=nmax: for k from 0 to kmax do A:= proc(x): add(a(n)*x^n, n=0..k) end: f(x):=series(1/(1 + x*A(x^6)),x,k+1); for n from 0 to k do x(n):=coeff(f(x),x,n) od: a(k):=x(k): od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Aug 08 2011

PARI

a(n)=local(A);A=1-x;for(i=1,n\6+1, A=1/(1+x*subst(A,x,x^6)+x*O(x^n)));polcoeff(A,n,x)

PARI

a(n)=local(M=contfracpnqn(concat(1, vector(ceil(log(n+1)/log(6))+1,n,1/x^(6^(n-1)))))); polcoeff(M[1,1]/M[2,1]+x*O(x^(7*n+1)),7*n+1)

Formula

a(0) = 1; a(n) = -Sum_{k=0..floor((n-1)/6)} a(k) * a(n-6*k-1). - Ilya Gutkovskiy, Mar 01 2022