A015224 Even pentagonal pyramidal numbers.

0, 6, 18, 40, 126, 196, 288, 550, 726, 936, 1470, 1800, 2176, 3078, 3610, 4200, 5566, 6348, 7200, 9126, 10206, 11368, 13950, 15376, 16896, 20230, 22050, 23976, 28158, 30420, 32800, 37926, 40678, 43560, 49726, 53016, 56448, 63750, 67626, 71656

Offset: 0

Author

Mohammad K. Azarian

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Eric Weisstein's World of Mathematics, Pentagonal Pyramidal Number
• Index entries for linear recurrences with constant coefficients, signature (1, 0, 3, -3, 0, -3, 3, 0, 1, -1).

Crossrefs

Cf. A002411, A014800, A015223, A014799, A006752.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(2*x*(3+6*x+11*x^2+34*x^3+17*x^4+13*x^5+11*x^6+x^7)/((1-x)^4*(1+x +x^2)^3))); // G. C. Greubel, Aug 24 2018

Mathematica

LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1},{0,6,18,40,126,196,288,550, 726,936},40] (* Ant King, Oct 19 2012 *)

PARI

x='x+O('x^30); concat([0], Vec(2*x*(3+6*x+11*x^2+34*x^3 +17*x^4 +13*x^5+11*x^6+x^7)/((1-x)^4*(1+x +x^2)^3))) \\ G. C. Greubel, Aug 24 2018

Formula

From Ant King, Oct 24 2012: (Start)

a(n) = a(n-1) +3*a(n-3) -3*a(n-4) -3*a(n-6) +3*a(n-7) +a(n-9) -a(n-10).

a(n) = 3*a(n-3) -3*a(n-6) +a(n-9) +192.

Sum_{n>=0} 1/a(n) = log(2)/2 + Pi/4 + 5*Pi^2/24 - 2 - C = 0.27217..., where C is Catalan's constant (A006752).

G.f.: 2*x*(3+6*x+11*x^2+34*x^3+17*x^4+13*x^5+11*x^6+x^7) / ((1-x)^4*(1+x +x^2)^3). (End)

a(n) = A002411(A004772(n+1)). - Bruno Berselli, Oct 24 2012

Extensions

More terms from Patrick De Geest, Jul 14 1999