A137501 The even numbers repeated, with alternating signs.

0, 0, 2, -2, 4, -4, 6, -6, 8, -8, 10, -10, 12, -12, 14, -14, 16, -16, 18, -18, 20, -20, 22, -22, 24, -24, 26, -26, 28, -28, 30, -30, 32, -32, 34, -34, 36, -36, 38, -38, 40, -40, 42, -42, 44, -44, 46, -46, 48, -48, 50, -50, 52, -52, 54, -54, 56, -56, 58, -58, 60, -60, 62, -62, 64, -64, 66, -66, 68, -68, 70, -70, 72, -72, 74

Offset: 0

Author

Carlos Alberto da Costa Filho (cacau_dacosta(AT)hotmail.com), Apr 22 2008

Keywords

• easy
• sign

Comments

The general formula for alternating sums of powers of even integers is in terms of the Swiss-Knife polynomials P(n,x) A153641 (P(n,1)-(-1)^k P(n,2k+1))/2. Here n=1 and k shifted one place, thus a(k) = (P(1,1)-(-1)^(k-1) P(1,2(k-1)+1))/2. - Peter Luschny, Jul 12 2009

With just one 0 at the beginning, this is a permutation of all the even integers. - Alonso del Arte, Jun 24 2012

Links

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

Crossrefs

Cf. A052928, A064455, A123684, A153641.

Programs

Maple

den:= n -> (n-1/2+1/2*(-1)^n)*(-1)^n: seq(den(n),n=-10..10);
a := n -> (1+(-1)^n*(2*n-1))/2; # Peter Luschny, Jul 12 2009

Mathematica

Flatten[Table[{2n, -2n}, {n, 0, 39}]] (* Alonso del Arte, Jun 24 2012 *)
With[{enos=2*Range[0,40]},Riffle[enos,-enos]] (* Harvey P. Dale, Oct 12 2014 *)

Formula

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

From R. J. Mathar, Feb 14 2010: (Start)

a(n) = -a(n-1) + a(n-2) + a(n-3).

G.f.: 2*x^2/((1-x) * (1+x)^2). (End)

a(n) = A064455(n) - A123684(n). - Jaroslav Krizek, Mar 22 2011