A110036 Constant terms of the partial quotients of the continued fraction expansion of 1 + Sum_{n>=0} 1/x^(2^n), where each partial quotient has the form {x + a(n)} after the initial constant term of 1.

1, -1, 2, 0, 0, -2, 0, 2, 0, -2, 2, 0, -2, 0, 0, 2, 0, -2, 2, 0, 0, -2, 0, 2, -2, 0, 2, 0, -2, 0, 0, 2, 0, -2, 2, 0, 0, -2, 0, 2, 0, -2, 2, 0, -2, 0, 0, 2, -2, 0, 2, 0, 0, -2, 0, 2, -2, 0, 2, 0, -2, 0, 0, 2, 0, -2, 2, 0, 0, -2, 0, 2, 0, -2, 2, 0, -2, 0, 0, 2, 0, -2, 2, 0, 0, -2, 0, 2, -2, 0, 2, 0, -2, 0, 0, 2, -2, 0, 2, 0, 0, -2, 0, 2, 0, -2, 2, 0, -2, 0, 0, 2, -2, 0

Offset: 0

Author

Paul D. Hanna, Jul 08 2005

Keywords

• cofr
• sign

Comments

Suggested by Ralf Stephan.

For n>1, |a(n)| = 2*A090678(n) where A090678(n) = A088567(n) mod 2 and A088567(n) = number of "non-squashing" partitions of n into distinct parts.

Examples

			1 + 1/x + 1/x^2 + 1/x^4 + 1/x^8 + 1/x^16 + ... =
[1; x - 1, x + 2, x, x, x - 2, x, x + 2, x, x - 2, ...].
		

Links

• Paul Barry, Conjectures and results on some generalized Rueppel sequences, arXiv:2107.00442 [math.CO], 2021.

Crossrefs

Cf. A090678, A088567.

Programs

PARI

contfrac(1+sum(n=0,10,1/x^(2^n)))

PARI

a(n)=polcoeff((1-x+3*x^2+x^3)/(1+x^2)- 2*sum(k=1,#binary(n),x^(3*2^(k-1))/prod(j=0,k,1+x^(2^j)+x*O(x^n))),n)

PARI

a(n)=subst(contfrac(1+sum(k=0,#binary(n+1),1/x^(2^k)))[n+1],x,0)

Formula

G.f. (1-x+3*x^2+x^3)/(1+x^2) - 2*Sum_{k>=1} x^(3*2^(k-1))/Product_{j=0..k} (1+x^(2^j)).