A144314 a(n) = 3*n*(6*n + 1).

0, 21, 78, 171, 300, 465, 666, 903, 1176, 1485, 1830, 2211, 2628, 3081, 3570, 4095, 4656, 5253, 5886, 6555, 7260, 8001, 8778, 9591, 10440, 11325, 12246, 13203, 14196, 15225, 16290, 17391, 18528, 19701, 20910, 22155, 23436, 24753, 26106, 27495

Offset: 0

Author

Reinhard Zumkeller, Sep 17 2008

Keywords

• nonn
• easy

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A014105, A033585, A049453, A081266, A144312.

Programs

Magma

[18*n^2+3*n: n in [0..50]]; // Vincenzo Librandi, Dec 18 2014

Maple

A144314:=n->3*n*(6*n+1): seq(A144314(n), n=0..70); # Wesley Ivan Hurt, Dec 16 2015

Mathematica

Table[3n(6n+1),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,21,78},40] (* Harvey P. Dale, Dec 17 2014 *)
CoefficientList[Series[x (21 + 15 x) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 18 2014 *)

PARI

a(n)=3*n*(6*n+1) \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = A000217(6*n) = A014105(3*n) = A081266(2*n).

G.f.: 3*x*(7+5*x)/(1-x)^3. - Vincenzo Librandi, Dec 18 2014

From Wesley Ivan Hurt, Dec 16 2015: (Start)

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

a(n) = 3 * A049453(n). (End)

E.g.f.: 3*exp(x)*x*(7 + 6*x). - Stefano Spezia, Jun 29 2021

From Amiram Eldar, May 11 2025: (Start)

Sum_{n>=1} 1/a(n) = 2 - Pi/(2*sqrt(3)) - 2*log(2)/3 - log(3)/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/3 + log(2)/3 + log(2+sqrt(3))/sqrt(3) - 2. (End)