A374157 a(n) = (-1)^floor(n/2)*n.

0, 1, -2, -3, 4, 5, -6, -7, 8, 9, -10, -11, 12, 13, -14, -15, 16, 17, -18, -19, 20, 21, -22, -23, 24, 25, -26, -27, 28, 29, -30, -31, 32, 33, -34, -35, 36, 37, -38, -39, 40, 41, -42, -43, 44, 45, -46, -47, 48, 49, -50, -51, 52, 53, -54, -55, 56, 57, -58, -59

Offset: 0

Author

Peter Luschny, Jun 30 2024

Keywords

• sign
• easy

Comments

For all odd numbers n (A005408) and all whole numbers z (A001057) K(z/n) = K(a(n)/z), where K(z/n) denotes the Kronecker symbol (A372728). This fact is equivalent to the law of quadratic reciprocity and its first and second supplement.

Links

• Index entries for linear recurrences with constant coefficients, signature (0,-2,0,-1).

Crossrefs

Cf. A005408, A001057, A057077, A196521, A372728.

Cf. A001477, A181983.

Programs

Maple

a := n -> (-1)^iquo(n, 2)*n: seq(a(n), n = 0..59);

Mathematica

Array[(-1)^Floor[#/2]*# &, 60, 0] (* Michael De Vlieger, Jun 30 2024 *)

PARI

a(n) = (-1)^(n\2) * n; \\ Amiram Eldar, Jun 30 2024

Python

def A374157(n): return (-1)**(n // 2)*n

Python

def A374157(n): return -n if n&2 else n # Chai Wah Wu, Jun 30 2024

Formula

Sum_{n>=1} 1/a(n) = Pi/4 - log(2)/2 = A196521.

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

a(n) = n! * [x^n] x*(cos(x) - sin(x)). - Stefano Spezia, Jun 30 2024

a(n) = n*A057077(n). - Michel Marcus, Jul 01 2024