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A372583
a(n) = (3*n^5 + 5*n^3)/8.
4
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
1,2
COMMENTS
Sum of pentagonal numbers in increasing groups 1, 5+12, 22+35+51, 70+92+117+145 etc.
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)
EXAMPLE
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.
MATHEMATICA
A372583[n_] := (3*n^5 + 5*n^3)/8; Array[A372583, 50] (* _Paolo Xausa_, May 25 2024 *)
CROSSREFS
Cf. A260513 (for triangular numbers), A072474 (for squares).
Cf. A000326 (pentagonal numbers), A002411 (their partial sums).
Sequence in context: A159031 A080441 A135400 * A052254 A156851 A354183
KEYWORD
nonn,easy
AUTHOR
_Kelvin Voskuijl_, May 05 2024
STATUS
approved