A115638 - OEIS

%I #19 Jul 18 2024 04:40:08

%S 1,-1,5,-1,-3,-1,21,-1,-3,-1,-11,-1,-3,-1,85,-1,-3,-1,-11,-1,-3,-1,

%T -43,-1,-3,-1,-11,-1,-3,-1,341,-1,-3,-1,-11,-1,-3,-1,-43,-1,-3,-1,-11,

%U -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

%N A Jacobsthal-related divide-and-conquer sequence.

%C Partial sums are A115637.

%H Antti Karttunen, <a href="/A115638/b115638.txt">Table of n, a(n) for n = 0..1024</a>

%H R. Stephan, <a href="https://arxiv.org/abs/math/0307027">Divide-and-conquer generating functions. I. Elementary sequences</a>, arXiv:math/0307027 [math.CO], 2003.

%F 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).

%F a(0) = 1; for n >= 1, a(n) = A115637(n) - A115637(n-1). - _Antti Karttunen_, Nov 02 2018

%t A115637[n_] := FromDigits[1 - IntegerDigits[n + 2, 2], 4];

%t Differences[Array[A115637, 100, -1]] (* _Paolo Xausa_, Jul 17 2024 *)

%o (PARI) A115638(n) = if(!n,1, A115637(n)-A115637(n-1)); \\ (Needs also code from A115637) - _Antti Karttunen_, Nov 02 2018

%o (Python)

%o 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

%Y Cf. A001045, A115637.

%K easy,sign

%O 0,3

%A _Paul Barry_, Jan 27 2006