A268351 a(n) = 3*n*(9*n - 1)/2.

0, 12, 51, 117, 210, 330, 477, 651, 852, 1080, 1335, 1617, 1926, 2262, 2625, 3015, 3432, 3876, 4347, 4845, 5370, 5922, 6501, 7107, 7740, 8400, 9087, 9801, 10542, 11310, 12105, 12927, 13776, 14652, 15555, 16485, 17442, 18426, 19437, 20475, 21540, 22632, 23751, 24897, 26070, 27270

Offset: 0

Author

Ilya Gutkovskiy, Feb 02 2016

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000326, A001318, A016766, A022266, A022284, A081266, A188623, A211538.

Programs

Magma

[3*n*(9*n-1)/2: n in [0..50]]; // Vincenzo Librandi, Feb 04 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]

PARI

a(n)=3*n*(9*n-1)/2 \\ Charles R Greathouse IV, Jul 26 2016

Formula

G.f.: 3*x*(4 + 5*x)/(1 - x)^3.

a(n) = binomial(9*n,2)/3.

a(n) = A000326(3*n) = 3*A022266(n).

a(n) = A211538(6*n+2).

a(n) = A001318(6*n-1), with A001318(-1)=0.

a(n) = A188623(9*n-2), with A188623(-2)=0.

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]

From Elmo R. Oliveira, Dec 27 2024: (Start)

E.g.f.: 3*exp(x)*x*(8 + 9*x)/2.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.

a(n) = A022284(n) - n. (End)

Extensions

Edited by Bruno Berselli, Feb 03 2016