A145919 A000332(n) = a(n)*(3*a(n) - 1)/2.

0, 0, 0, 0, 1, 2, -3, 5, 7, -9, 12, 15, -18, 22, 26, -30, 35, 40, -45, 51, 57, -63, 70, 77, -84, 92, 100, -108, 117, 126, -135, 145, 155, -165, 176, 187, -198, 210, 222, -234, 247, 260, -273, 287, 301, -315, 330, 345, -360, 376, 392, -408, 425, 442, -459, 477

Offset: 0

Author

Matthew Vandermast, Oct 28 2008

Keywords

• easy
• sign

Comments

As the formula in the description shows, all members of A000332 belong to the generalized pentagonal sequence (A001318). A001318 also lists all nonnegative numbers that belong to A145919.

Examples

			a(6) = -3 and A000332(6) = (-3)(-10)/2 = 15.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Pentagonal Number.
• Eric Weisstein's World of Mathematics, Pentatope Number.
• Index entries for linear recurrences with constant coefficients, signature (0,0,3,0,0,-3,0,0,1).

Crossrefs

Cf. A000326, A145920.

Programs

Mathematica

CoefficientList[Series[(-x^4*(x^4 + 2*x^3 - 3*x^2 + 2*x + 1))/((x - 1)^3*(1 + x^2 + x)^3), {x,0,50}], x] (* G. C. Greubel, Jun 13 2017 *)
LinearRecurrence[{0,0,3,0,0,-3,0,0,1},{0,0,0,0,1,2,-3,5,7},60] (* Harvey P. Dale, Feb 13 2023 *)

PARI

my(x='x+O('x^50)); concat([0, 0, 0, 0], Vec((-x^4*(x^4 +2*x^3 -3*x^2 +2*x +1))/((x-1)^3*(1+x^2+x)^3))) \\ G. C. Greubel, Jun 13 2017

Formula

a(n+3) = A001840(n) when 3 does not divide n, A001840(n)*-1 otherwise.

After first two zeros, this sequence consists of all values of A001318(n) and A045943(n)*(-1), n>=0, sorted in order of increasing absolute value.

G.f.: (-x^4*(x^4+2*x^3-3*x^2+2*x+1))/((x-1)^3*(1+x^2+x)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009

a(n) = ((n^2-3*n+1)*(1-4*cos(2*Pi*n/3))+3)/18. - Natalia L. Skirrow, Apr 14 2025