OFFSET
0,2
COMMENTS
First trisection of pentagonal numbers (A000326).
More generally, the ordinary generating function for the first trisection of k-gonal numbers is 3*x*(k - 1 + (2*k - 5)*x)/(1 - x)^3.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pentagonal Number
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
G.f.: 3*x*(4 + 5*x)/(1 - x)^3.
a(n) = binomial(9*n,2)/3.
a(n) = A211538(6*n+2).
Sum_{n>=1} 1/a(n) = 0.132848490245209886617568... = (-Pi*cot(Pi/9) + 5*log(3) + 4*cos(Pi/9)*log(cos(Pi/18)) - 4*cos(2*Pi/9)*log(sin(Pi/9)) - 4*log(sin(2*Pi/9))*sin(Pi/18))/3. [Corrected by Vaclav Kotesovec, Feb 25 2016]
MATHEMATICA
Table[3 n (9 n - 1)/2, {n, 0, 45}]
Table[Binomial[9 n, 2]/3, {n, 0, 45}]
LinearRecurrence[{3, -3, 1}, {0, 12, 51}, 45]
PROG
(Magma) [3*n*(9*n-1)/2: n in [0..50]]; // Vincenzo Librandi, Feb 04 2016
(PARI) a(n)=3*n*(9*n-1)/2 \\ Charles R Greathouse IV, Jul 26 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Feb 02 2016
EXTENSIONS
Edited by Bruno Berselli, Feb 03 2016
STATUS
approved