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

0,3

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.

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A001045, A115637.

Sequence in context: A348498 A206076 A329374 * A342375 A055515 A363437

Adjacent sequences: A115635 A115636 A115637 * A115639 A115640 A115641

Keyword

easy,sign

Author

Paul Barry, Jan 27 2006

Status

approved