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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 (list; graph; refs; listen; history; text; internal format)
 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 A271252 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

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Last modified October 19 04:40 EDT 2019. Contains 328211 sequences. (Running on oeis4.)