A363524 a(n) = 0 if 4 divides n + 1, otherwise (-1)^floor((n + 1) / 4) * 2^floor(n / 2).

1, 1, 2, 0, -4, -4, -8, 0, 16, 16, 32, 0, -64, -64, -128, 0, 256, 256, 512, 0, -1024, -1024, -2048, 0, 4096, 4096, 8192, 0, -16384, -16384, -32768, 0, 65536, 65536, 131072, 0, -262144, -262144, -524288, 0, 1048576, 1048576, 2097152, 0, -4194304, -4194304

Offset: 0

Author

Peter Luschny, Jun 17 2023

Keywords

• sign
• easy

Comments

We call chen(n) = 1 / a(n) if a(n) != 0 and otherwise 0, the Chen sequence, after Kwang-Wu Chen (see link). Note how this sequence is used in the computation of the Swiss-Knife polynomials A153641.

Links

• Winston de Greef, Table of n, a(n) for n = 0..6603
• Kwang-Wu Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,-4).

Crossrefs

Cf. A153641, A016116, A046978.

Programs

Maple

a := n -> if irem(n + 1, 4) = 0 then 0 else (-1)^iquo(n + 1, 4) * 2^iquo(n, 2) fi:
seq(a(n), n = 0..49);
# Alternative:
gf := (2*x^2 + x + 1)/(4*x^4 + 1): ser := series(gf, x, 24):
seq(coeff(ser, x, n), n = 0..20);

Mathematica

A363524list[nmax_]:=LinearRecurrence[{0,0,0,-4},{1,1,2,0},nmax+1];A363524list[100] (* Paolo Xausa, Aug 06 2023 *)

PARI

a(n)=if(n % 4 == 3, 0, (-1)^((n + 1) \ 4) * 2^(n \ 2)) \\ Winston de Greef, Jun 30 2023

SageMath

def a(n): return 0 if 4.divides(n + 1) else (-1)^((n + 1) // 4) * 2^(n // 2)
print([a(n) for n in range(45)])

Formula

a(n) = [x^n] (2*x^2 + x + 1) / (4*x^4 + 1).

a(n) = A016116(n) * A046978(n + 1).