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

0,3

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.

Links

Table of n, a(n) for n=0..113.

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

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)).

Example

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, ...].

Prog

(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)

Crossrefs

Cf. A090678, A088567.

Sequence in context: A029305 A339441 A226914 * A308046 A289323 A086937

Adjacent sequences: A110033 A110034 A110035 * A110037 A110038 A110039

Keyword

cofr,sign

Author

Paul D. Hanna, Jul 08 2005

Status

approved