A266088 Alternating sum of 12-gonal (or dodecagonal) numbers.

0, -1, 11, -22, 42, -63, 93, -124, 164, -205, 255, -306, 366, -427, 497, -568, 648, -729, 819, -910, 1010, -1111, 1221, -1332, 1452, -1573, 1703, -1834, 1974, -2115, 2265, -2416, 2576, -2737, 2907, -3078, 3258, -3439, 3629, -3820, 4020, -4221, 4431, -4642

Offset: 0

Author

Ilya Gutkovskiy, Dec 21 2015

Keywords

• sign
• easy

Comments

More generally, the ordinary generating function for the alternating sum of k-gonal numbers is -x*(1 - (k - 3)*x)/((1 - x)*(1 + x)^3).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• OEIS Wiki, Figurate numbers
• Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Crossrefs

Cf. A006578, A007587, A035608, A051624, A083392, A089594.

Programs

Magma

[1+(-1)^n*(5*n^2+n-2)/2: n in [0..50]]; // Vincenzo Librandi, Dec 21 2015

Mathematica

Table[1 + (-1)^n (5 n^2 + n - 2)/2, {n, 0, 43}]
CoefficientList[Series[-x (1 - 9 x)/((1 - x) (1 + x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *)

PARI

x='x+O('x^100); concat(0, Vec(-x*(1-9*x)/((1-x)*(1+x)^3))) \\ Altug Alkan, Dec 21 2015

Formula

G.f.: -x*(1 - 9*x)/((1 - x)*(1 + x)^3).

a(n) = 1 + (-1)^n*(5*n^2 + n - 2)/2.

a(n) = Sum_{k = 0..n} (-1)^k*A051624(k).

Lim_{n -> infinity} a(n + 1)/a(n) = -1.