A194454 a(n) = 12*n^2 + 2*n + 1.

1, 15, 53, 115, 201, 311, 445, 603, 785, 991, 1221, 1475, 1753, 2055, 2381, 2731, 3105, 3503, 3925, 4371, 4841, 5335, 5853, 6395, 6961, 7551, 8165, 8803, 9465, 10151, 10861, 11595, 12353, 13135, 13941, 14771, 15625, 16503, 17405, 18331, 19281

Offset: 0

Author

Bruno Berselli, Aug 24 2011

Keywords

• nonn
• easy

Comments

A142241 gives the first differences.

Inverse binomial transform of this sequence: 1, 14, 24, 0, 0 (0 continued).

a(n)*a(n-1)-11 is a square, precisely 4*A051866(n)^2.

Sequence found by reading the line from 1, in the direction 1, 15, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012

Examples

			Using these numbers we can write:
  1, 15, 53, 115, 201, 311, 445,  603,  785,  991, 1221, ...
  0,  0,  1,  15,  53, 115, 201,  311,  445,  603,  785, ...
  0,  0,  0,   0,   1,  15,  53,  115,  201,  311,  445, ...
  0,  0,  0,   0,   0,   0,   1,   15,   53,  115,  201, ...
  0,  0,  0,   0,   0,   0,   0,    0,    1,   15,   53, ...
  0,  0,  0,   0,   0,   0,   0,    0,    0,    0,    1, ...
  ======================================================
  The sums of the columns give the sequence A172073 (after 0):
  1, 15, 54, 130, 255, 441, 700, 1044, 1485, 2035, 2706, ...
		

Links

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

Crossrefs

Cf. A154106, A172073, A049453.

Programs

Magma

[12*n^2+2*n+1: n in [0..40]];

Mathematica

Table[12 n^2 + 2 n + 1, {n, 0, 50}] (* Vincenzo Librandi, Mar 26 2013 *)

PARI

for(n=0, 40, print1(12*n^2+2*n+1", "));

Formula

G.f.: (1+x)*(1+11*x)/(1-x)^3.

a(n) = A154106(-n-1).

a(n) = 2*A049453(n) + 1.

a(n) = A051866(n) + A051866(n+1). - Charlie Marion, Nov 15 2019

E.g.f.: exp(x)*(1 + 14*x + 12*x^2). - Stefano Spezia, Nov 15 2019