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A129863
Sums of three consecutive pentagonal numbers.
5
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?
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
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, May 23 2007, May 24 2007
EXTENSIONS
Offset corrected by Eric Rowland, Aug 15 2017
STATUS
approved