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

Offset

0,1

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?

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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

Formula

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

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

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

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

Example

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

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 *)

Prog

(PARI) a(n)=n*(9*n+15)/2+6 \\ Charles R Greathouse IV, Jun 17 2017
(Magma) [(9/2)*(n^2)+(15/2)*n+6: n in [0..50]]; // Vincenzo Librandi, Aug 16 2017

Crossrefs

Cf. A000326, A007667, A034961, A129803.

Sequence in context: A270081 A261651 A270215 * A272014 A272453 A370349

Adjacent sequences: A129860 A129861 A129862 * A129864 A129865 A129866

Keyword

easy,nonn

Author

Jonathan Vos Post, May 23 2007, May 24 2007

Extensions

Offset corrected by Eric Rowland, Aug 15 2017

Status

approved