# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a129863 Showing 1-1 of 1 %I A129863 #27 Nov 16 2024 16:57:06 %S A129863 6,18,39,69,108,156,213,279,354,438,531,633,744,864,993,1131,1278, %T A129863 1434,1599,1773,1956,2148,2349,2559,2778,3006,3243,3489,3744,4008, %U A129863 4281,4563,4854,5154,5463,5781,6108,6444,6789,7143,7506,7878,8259,8649,9048,9456,9873 %N A129863 Sums of three consecutive pentagonal numbers. %C A129863 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? %H A129863 Vincenzo Librandi, Table of n, a(n) for n = 0..1000 %H A129863 Index entries for linear recurrences with constant coefficients, signature (3,-3,1). %F A129863 a(n) = P(n) + P(n+1) + P(n+2) where P(n) = A000326(n) = n*(3*n-1)/2. %F A129863 a(n) = (9/2)*(n^2) + (15/2)*n + 6. %F A129863 a(n) = (3*n^2 + 5*n + 4)*(3/2). - _Stefan Steinerberger_, May 27 2007 %F A129863 G.f.: 3*(2+x^2)/(1-x)^3. - _Colin Barker_, Feb 13 2012 %F A129863 From _Elmo R. Oliveira_, Nov 16 2024: (Start) %F A129863 E.g.f.: 3*exp(x)*(3*x^2 + 8*x + 4)/2. %F A129863 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End) %e A129863 a(0) = 6 = A000326(0) + A000326(1) + A000326(2) = 0 + 1 + 5. %e A129863 a(1) = 18 = A000326(1) + A000326(2) + A000326(3) = 1 + 5 + 12. %t A129863 Table[(3/2)*(4 + 5*n + 3*n^2), {n, 0, 100}] (* _Stefan Steinerberger_, May 27 2007 *) %t A129863 CoefficientList[Series[3 (2 + x^2) / (1 - x)^3, {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 16 2017 *) %t A129863 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 *) %o A129863 (PARI) a(n)=n*(9*n+15)/2+6 \\ _Charles R Greathouse IV_, Jun 17 2017 %o A129863 (Magma) [(9/2)*(n^2)+(15/2)*n+6: n in [0..50]]; // _Vincenzo Librandi_, Aug 16 2017 %Y A129863 Cf. A000326, A007667, A034961, A129803. %K A129863 easy,nonn,changed %O A129863 0,1 %A A129863 _Jonathan Vos Post_, May 23 2007, May 24 2007 %E A129863 Offset corrected by _Eric Rowland_, Aug 15 2017 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE