A269429 Alternating sum of octagonal pyramidal numbers.

0, -1, 8, -22, 48, -87, 144, -220, 320, -445, 600, -786, 1008, -1267, 1568, -1912, 2304, -2745, 3240, -3790, 4400, -5071, 5808, -6612, 7488, -8437, 9464, -10570, 11760, -13035, 14400, -15856, 17408, -19057, 20808, -22662, 24624, -26695, 28880, -31180, 33600

Offset

0,3

Links

Table of n, a(n) for n=0..40.

OEIS Wiki, Figurate numbers

Eric Weisstein's World of Mathematics, Pyramidal Number

Index entries for linear recurrences with constant coefficients, signature (-3,-2,2,3,1).

Formula

G.f.: x*(1 - 5*x)/((x - 1)*(x + 1)^4).

a(n) = ((2*n^3 + 4*n^2 - 1)*(-1)^n + 1)/4.

a(n) = Sum_{k = 0..n} (-1)^k*A002414(k).

Sum_{n>=1} 1/a(n) = -0.906890389180715042293808708467278316660747358... . - Vaclav Kotesovec, Feb 26 2016

Mathematica

Table[((2 n^3 + 4 n^2 - 1) (-1)^n + 1)/4, {n, 0, 40}]
LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -1, 8, -22, 48}, 41]

Prog

(PARI) a(n)=(2*n^3 + 4*n^2 - 1)*(-1)^n\/4 \\ Charles R Greathouse IV, Jul 26 2016

Crossrefs

Cf. A000292, A002414, A002717, A173196, A266677.

Sequence in context: A189584 A239360 A211532 * A305181 A048489 A124701

Adjacent sequences: A269426 A269427 A269428 * A269430 A269431 A269432

Keyword

sign,easy

Author

Ilya Gutkovskiy, Feb 26 2016

Status

approved