A266086 Alternating sum of 9-gonal (or nonagonal) numbers.

0, -1, 8, -16, 30, -45, 66, -88, 116, -145, 180, -216, 258, -301, 350, -400, 456, -513, 576, -640, 710, -781, 858, -936, 1020, -1105, 1196, -1288, 1386, -1485, 1590, -1696, 1808, -1921, 2040, -2160, 2286, -2413, 2546, -2680, 2820, -2961, 3108, -3256, 3410

Offset: 0

Author

Ilya Gutkovskiy, Dec 21 2015

Keywords

• sign
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• OEIS Wiki, Figurate numbers
• Eric Weisstein's World of Mathematics, Nonagonal Number
• Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Crossrefs

Cf. A001106, A006578, A007584, A035608, A083392, A089594.

Programs

Magma

[(14*(-1)^n*n^2 + 4*(-1)^n*n - 5*(-1)^n + 5)/8: n in [0..50]]; // Vincenzo Librandi, Dec 21 2015

Mathematica

Table[((14 n^2 + 4 n - 5) (-1)^n + 5)/8, {n, 0, 44}]
CoefficientList[Series[(x - 6 x^2)/(x^4 + 2 x^3 - 2 x - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *)

PARI

x='x+O('x^100); concat(0, Vec(-x*(1-6*x)/((1-x)*(1+x)^3))) \\ Altug Alkan, Dec 21 2015

Formula

G.f.: -x*(1 - 6*x)/((1 - x)*(1 + x)^3).

a(n) = ((14*n^2 + 4*n - 5)*(-1)^n + 5)/8.

a(n) = Sum_{k = 0..n} (-1)^k*A001106(k).

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