A016946 a(n) = (6*n+3)^2.

9, 81, 225, 441, 729, 1089, 1521, 2025, 2601, 3249, 3969, 4761, 5625, 6561, 7569, 8649, 9801, 11025, 12321, 13689, 15129, 16641, 18225, 19881, 21609, 23409, 25281, 27225, 29241, 31329, 33489, 35721, 38025, 40401, 42849, 45369, 47961, 50625, 53361, 56169

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..2000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000290, A002378, A006752, A016754, A016945, A086729.

Programs

Magma

[(6*n+3)^2: n in [0..60]]; // Vincenzo Librandi, May 05 2011

Maple

A016946:=n->(6*n+3)^2: seq(A016946(n), n=0..50); # Wesley Ivan Hurt, Oct 13 2014

Mathematica

(6 Range[0, 50] + 3)^2 (* or *)
CoefficientList[Series[9 (1 + 6 x + x^2)/(1 - x)^3, {x, 0, 50}], x] (* Wesley Ivan Hurt, Oct 13 2014 *)
LinearRecurrence[{3,-3,1},{9,81,225},40] (* Harvey P. Dale, Jul 13 2015 *)

PARI

a(n)=(6*n+3)^2 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 36*A002378(n)+9. - Jean-Bernard François, Oct 12 2014

From Wesley Ivan Hurt, Oct 13 2014: (Start)

G.f.: 9*(1+6*x+x^2)/(1-x)^3.

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

a(n) = A016945(n)^2 = A000290(A016945(n)). (End)

Sum_{n>=0} 1/a(n) = A086729. - Amiram Eldar, Nov 16 2020

a(n) = 9*A016754(n). - R. J. Mathar, Dec 11 2020

Sum_{n>=0} (-1)^n/a(n) = G/9, where G is Catalan's constant (A006752). - Amiram Eldar, Mar 30 2022

E.g.f.: 9*exp(x)*(1 + 8*x + 4*x^2). - Stefano Spezia, Aug 19 2022