A115638 A Jacobsthal-related divide-and-conquer sequence.

1, -1, 5, -1, -3, -1, 21, -1, -3, -1, -11, -1, -3, -1, 85, -1, -3, -1, -11, -1, -3, -1, -43, -1, -3, -1, -11, -1, -3, -1, 341, -1, -3, -1, -11, -1, -3, -1, -43, -1, -3, -1, -11, -1, -3, -1, -171, -1, -3, -1, -11, -1, -3, -1, -43, -1, -3, -1, -11, -1, -3, -1, 1365, -1, -3, -1, -11, -1, -3, -1, -43

Offset: 0

Author

Paul Barry, Jan 27 2006

Keywords

• easy
• sign

Comments

Partial sums are A115637.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..1024
• R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.

Crossrefs

Cf. A001045, A115637.

Programs

Mathematica

A115637[n_] := FromDigits[1 - IntegerDigits[n + 2, 2], 4];
Differences[Array[A115637, 100, -1]] (* Paolo Xausa, Jul 17 2024 *)

PARI

A115638(n) = if(!n,1, A115637(n)-A115637(n-1)); \\ (Needs also code from A115637) - Antti Karttunen, Nov 02 2018

Python

def A115638(n): return int(bin((~(n+2))^(-1<<(n+2).bit_length()))[2:],4)-int(bin((~(n+1))^(-1<<(n+1).bit_length()))[2:],4) # Chai Wah Wu, Jul 17 2024

Formula

G.f.: Sum_{k>=0} 4^k*x^(2^(k+1)-2)/(1+x^(2^k)); the g.f. G(x) satisfies G(x) - 4*x^2*G(x^2) = 1/(1+x).

a(0) = 1; for n >= 1, a(n) = A115637(n) - A115637(n-1). - Antti Karttunen, Nov 02 2018