A129863 Sums of three consecutive pentagonal numbers.

6, 18, 39, 69, 108, 156, 213, 279, 354, 438, 531, 633, 744, 864, 993, 1131, 1278, 1434, 1599, 1773, 1956, 2148, 2349, 2559, 2778, 3006, 3243, 3489, 3744, 4008, 4281, 4563, 4854, 5154, 5463, 5781, 6108, 6444, 6789, 7143, 7506, 7878, 8259, 8649, 9048, 9456, 9873

Offset: 0

Author

Jonathan Vos Post, May 23 2007, May 24 2007

Keywords

• easy
• nonn

Comments

Arises in pentagonal number analog to A129803, Triangular numbers that are the sum of three consecutive triangular numbers. What are the pentagonal numbers which are the sum of three consecutive pentagonal numbers?

Examples

			a(0) = 6 = A000326(0) + A000326(1) + A000326(2) = 0 + 1 + 5.
a(1) = 18 = A000326(1) + A000326(2) + A000326(3) = 1 + 5 + 12.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000326, A007667, A034961, A129803.

Programs

Magma

[(9/2)*(n^2)+(15/2)*n+6: n in [0..50]]; // Vincenzo Librandi, Aug 16 2017

Mathematica

Table[(3/2)*(4 + 5*n + 3*n^2), {n, 0, 100}] (* Stefan Steinerberger, May 27 2007 *)
CoefficientList[Series[3 (2 + x^2) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 16 2017 *)
Total/@Partition[PolygonalNumber[5,Range[0,50]],3,1] (* Requires Mathematica version 10 or later *) (* or *) LinearRecurrence[{3,-3,1},{6,18,39},50] (* Harvey P. Dale, Nov 22 2018 *)

PARI

a(n)=n*(9*n+15)/2+6 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = P(n) + P(n+1) + P(n+2) where P(n) = A000326(n) = n*(3*n-1)/2.

a(n) = (9/2)*(n^2) + (15/2)*n + 6.

a(n) = (3*n^2 + 5*n + 4)*(3/2). - Stefan Steinerberger, May 27 2007

G.f.: 3*(2+x^2)/(1-x)^3. - Colin Barker, Feb 13 2012

From Elmo R. Oliveira, Nov 16 2024: (Start)

E.g.f.: 3*exp(x)*(3*x^2 + 8*x + 4)/2.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

Extensions

Offset corrected by Eric Rowland, Aug 15 2017