OFFSET
0,2
COMMENTS
First bisection of A051866.
More generally, the ordinary generating function for the even k-gonal numbers with even k or for the first bisection of k-gonal numbers, is (k*x + (3*k - 8)*x^2)/(1 - x )^3.
LINKS
OEIS Wiki, Figurate numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1)
FORMULA
G.f.: 2*x*(7 + 17*x)/(1 - x)^3.
E.g.f.: 2*exp(x)*x*(7 + 12*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*n*(12*n - 5).
Sum_{n>=1} 1/a(n) = (Pi - sqrt(3)*Pi + sqrt(3)*log(27) + sqrt(3)*log(64) + log(1728) + 6*log(sqrt(3)-1) + 2*sqrt(3)*log(sqrt(3)-1) - 6*log(sqrt(3)+1) - 2*sqrt(3)*log(sqrt(3)+1))/(20 + 20*sqrt(3)) = 0.102542837854…
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {0, 14, 76}, 41]
Table[2 n (12 n - 5), {n, 0, 40}]
PolygonalNumber[14, Range[0, 80, 2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 12 2017 *)
PROG
(PARI) concat(0, Vec(2*x*(7 + 17*x)/(1 - x)^3 + O(x^60))) \\ Michel Marcus, Mar 22 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Mar 22 2016
STATUS
approved