A372583 a(n) = (3*n^5 + 5*n^3)/8.

1, 17, 108, 424, 1250, 3051, 6517, 12608, 22599, 38125, 61226, 94392, 140608, 203399, 286875, 395776, 535517, 712233, 932824, 1205000, 1537326, 1939267, 2421233, 2994624, 3671875, 4466501, 5393142, 6467608, 7706924, 9129375, 10754551, 12603392, 14698233

Offset: 1

Author

Kelvin Voskuijl, May 05 2024

Keywords

• nonn
• easy

Comments

Sum of pentagonal numbers in increasing groups 1, 5+12, 22+35+51, 70+92+117+145 etc.

Examples

			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.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A260513 (for triangular numbers), A072474 (for squares).

Cf. A000326 (pentagonal numbers), A002411 (their partial sums).

Programs

Mathematica

A372583[n_] := (3*n^5 + 5*n^3)/8; Array[A372583, 50] (* Paolo Xausa, May 25 2024 *)

Formula

From Stefano Spezia, May 06 2024: (Start)

G.f.: x*(1 + 11*x + 21*x^2 + 11*x^3 + x^4)/(1 - x)^6.

E.g.f.: exp(x)*x*(8 + 60*x + 80*x^2 + 30*x^3 + 3*x^4)/8. (End)