A016778 a(n) = (3*n+1)^2.

1, 16, 49, 100, 169, 256, 361, 484, 625, 784, 961, 1156, 1369, 1600, 1849, 2116, 2401, 2704, 3025, 3364, 3721, 4096, 4489, 4900, 5329, 5776, 6241, 6724, 7225, 7744, 8281, 8836, 9409, 10000, 10609, 11236, 11881, 12544, 13225, 13924, 14641, 15376, 16129, 16900, 17689

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

From Paul Curtz, Mar 28 2019: (Start)

Sequence is a spoke of the hexagonal spiral built from the terms of A016777:

.

\

100--97--94--91

\ \

49--46--43 88

/ \ \ \

52 16--13 40 85

/ / \ \ \ \

55 19 1 10 37 82

/ / / / / /

58 22 4---7 34 79

\ \ / /

61 25--28--31 76

\ /

64--67--70--73

(End)

Links

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

Crossrefs

Cf. A000212, A000290, A016777, A016779, A016780, A016781, A214550, A247792, A262178.

Programs

Mathematica

(3*Range[0,50]+1)^2 (* or *) LinearRecurrence[{3,-3,1},{1,16,49},50] (* Harvey P. Dale, Mar 03 2013 *)

Maxima

A016778(n):=(3*n+1)^2$
makelist(A016778(n),n,0,20); /* Martin Ettl, Nov 12 2012 */

PARI

a(n)=(3*n+1)^2 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = a(n-1) + 3*(6*n-1); a(0)=1. - Vincenzo Librandi, Nov 20 2010

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=16, a(2)=49. - Harvey P. Dale, Mar 03 2013

a(n) = A247792(n) + 6*n. - Miquel Cerda, Oct 23 2016

G.f.: (1 + 13*x + 4*x^2)/(1 - x)^3. - Ilya Gutkovskiy, Oct 23 2016

a(n) = A000212(3*n) + A000212(1+3*n) + A000212(2+3*n). - Paul Curtz, Mar 28 2019

From Amiram Eldar, Nov 12 2020: (Start)

Sum_{n>=0} 1/a(n) = A214550.

Sum_{n>=0} (-1)^n/a(n) = A262178. (End)

From Elmo R. Oliveira, May 29 2025: (Start)

E.g.f.: exp(x)*(1 + 15*x + 9*x^2).

a(n) = A000290(A016777(n)) = A016777(n)^2. (End)