OFFSET
1,2
COMMENTS
Sum of pentagonal numbers in increasing groups 1, 5+12, 22+35+51, 70+92+117+145 etc.
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).
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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Kelvin Voskuijl, May 05 2024
STATUS
approved