This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A372583 #33 May 25 2024 15:32:12 %S A372583 1,17,108,424,1250,3051,6517,12608,22599,38125,61226,94392,140608, %T A372583 203399,286875,395776,535517,712233,932824,1205000,1537326,1939267, %U A372583 2421233,2994624,3671875,4466501,5393142,6467608,7706924,9129375,10754551,12603392,14698233 %N A372583 a(n) = (3*n^5 + 5*n^3)/8. %C A372583 Sum of pentagonal numbers in increasing groups 1, 5+12, 22+35+51, 70+92+117+145 etc. %H A372583 Paolo Xausa, <a href="/A372583/b372583.txt">Table of n, a(n) for n = 1..10000</a> %H A372583 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1). %F A372583 From _Stefano Spezia_, May 06 2024: (Start) %F A372583 G.f.: x*(1 + 11*x + 21*x^2 + 11*x^3 + x^4)/(1 - x)^6. %F A372583 E.g.f.: exp(x)*x*(8 + 60*x + 80*x^2 + 30*x^3 + 3*x^4)/8. (End) %e A372583 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 A372583[n_] := (3*n^5 + 5*n^3)/8; Array[A372583, 50] (* _Paolo Xausa_, May 25 2024 *) %Y A372583 Cf. A260513 (for triangular numbers), A072474 (for squares). %Y A372583 Cf. A000326 (pentagonal numbers), A002411 (their partial sums). %K A372583 nonn,easy %O A372583 1,2 %A A372583 _Kelvin Voskuijl_, May 05 2024