%I #33 May 25 2024 15:32:12
%S 1,17,108,424,1250,3051,6517,12608,22599,38125,61226,94392,140608,
%T 203399,286875,395776,535517,712233,932824,1205000,1537326,1939267,
%U 2421233,2994624,3671875,4466501,5393142,6467608,7706924,9129375,10754551,12603392,14698233
%N a(n) = (3*n^5 + 5*n^3)/8.
%C Sum of pentagonal numbers in increasing groups 1, 5+12, 22+35+51, 70+92+117+145 etc.
%H Paolo Xausa, <a href="/A372583/b372583.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F From _Stefano Spezia_, May 06 2024: (Start)
%F G.f.: x*(1 + 11*x + 21*x^2 + 11*x^3 + x^4)/(1 - x)^6.
%F E.g.f.: exp(x)*x*(8 + 60*x + 80*x^2 + 30*x^3 + 3*x^4)/8. (End)
%e The first ten pentagonal numbers are 1, 5, 12, 22, 35, 51, 70, 92, 117, and 145. Taking them in groups, respectively, of 1, 2, 3, and 4, i.e., (1), (5, 12), (22, 35, 51), and (70, 92, 117, 145), and summing each group separately gives 1, 17, 108, 424.
%t A372583[n_] := (3*n^5 + 5*n^3)/8; Array[A372583, 50] (* _Paolo Xausa_, May 25 2024 *)
%Y Cf. A260513 (for triangular numbers), A072474 (for squares).
%Y Cf. A000326 (pentagonal numbers), A002411 (their partial sums).
%K nonn,easy
%O 1,2
%A _Kelvin Voskuijl_, May 05 2024