A016766 a(n) = (3*n)^2.

0, 9, 36, 81, 144, 225, 324, 441, 576, 729, 900, 1089, 1296, 1521, 1764, 2025, 2304, 2601, 2916, 3249, 3600, 3969, 4356, 4761, 5184, 5625, 6084, 6561, 7056, 7569, 8100, 8649, 9216, 9801, 10404, 11025, 11664, 12321, 12996, 13689, 14400, 15129, 15876, 16641, 17424

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of edges of the complete tripartite graph of order 6n, K_n, n, 4n. - Roberto E. Martinez II, Jan 07 2002

Area of a square with side 3n. - Wesley Ivan Hurt, Sep 24 2014

Right-hand side of the binomial coefficient identity Sum_{k = 0..3*n} (-1)^(n+k+1)* binomial(3*n,k)*binomial(3*n + k,k)*(3*n - k) = a(n). - Peter Bala, Jan 12 2022

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..200
• John Elias, Illustration of initial terms.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Numbers of the form 9*n^2 + k*n, for integer n: this sequence (k = 0), A132355 (k = 2), A185039 (k = 4), A057780 (k = 6), A218864 (k = 8). - Jason Kimberley, Nov 09 2012

Cf. A000217, A000290, A008585, A033428, A051624, A086089, A195042.

Programs

Magma

[(3*n)^2 : n in [0..50]]; // Wesley Ivan Hurt, Sep 24 2014

Maple

A016766:=n->(3*n)^2: seq(A016766(n), n=0..50); # Wesley Ivan Hurt, Sep 24 2014

Mathematica

(3Range[0, 49])^2 (* Alonso del Arte, Sep 24 2014 *)

Maxima

A016766(n):=(3*n)^2$
makelist(A016766(n),n,0,20); /* Martin Ettl, Nov 12 2012 */

PARI

a(n)=9*n^2 \\ Charles R Greathouse IV, Sep 28 2015

Formula

a(n) = 9*n^2 = 9*A000290(n). - Omar E. Pol, Dec 11 2008

a(n) = 3*A033428(n). - Omar E. Pol, Dec 13 2008

a(n) = a(n-1) + 9*(2*n-1) for n > 0, a(0)=0. - Vincenzo Librandi, Nov 19 2010

From Wesley Ivan Hurt, Sep 24 2014: (Start)

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

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

a(n) = A000290(A008585(n)). (End)

From Amiram Eldar, Jan 25 2021: (Start)

Sum_{n>=1} 1/a(n) = Pi^2/54.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/108.

Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/3)/(Pi/3).

Product_{n>=1} (1 - 1/a(n)) = sinh(Pi/2)/(Pi/2) = 3*sqrt(3)/(2*Pi) (A086089). (End)

a(n) = A051624(n) + 8*A000217(n). In general, if P(k,n) = the k-th n-gonal number, then (k*n)^2 = P(k^2 + 3,n) + (k^2 - 1)*A000217(n). - Charlie Marion, Mar 09 2022

From Elmo R. Oliveira, Nov 30 2024: (Start)

E.g.f.: 9*x*(1 + x)*exp(x).

a(n) = n*A008591(n) = A195042(2*n). (End)

Extensions

More terms from Zerinvary Lajos, May 30 2006