A051865 13-gonal (or tridecagonal) numbers: a(n) = n*(11*n - 9)/2.

0, 1, 13, 36, 70, 115, 171, 238, 316, 405, 505, 616, 738, 871, 1015, 1170, 1336, 1513, 1701, 1900, 2110, 2331, 2563, 2806, 3060, 3325, 3601, 3888, 4186, 4495, 4815, 5146, 5488, 5841, 6205, 6580, 6966, 7363, 7771, 8190, 8620, 9061, 9513

Offset: 0

Author

N. J. A. Sloane, Dec 15 1999

Keywords

• nonn
• easy

Comments

Sequence found by reading the line from 0, in the direction 0, 13, ... and the parallel line from 1, in the direction 1, 36, ..., in the square spiral whose vertices are the generalized 13-gonal numbers A195313. - Omar E. Pol, Jul 18 2012

References

• Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 189.
• E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Index to sequences related to polygonal numbers
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Programs

Mathematica

CoefficientList[Series[x (1 + 10 x) / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 19 2013 *)
LinearRecurrence[{3,-3,1},{0,1,13},50] (* Harvey P. Dale, Jul 12 2014 *)
Table[n*(11*n - 9)/2, {n, 0, 100}] (* Robert Price, Oct 11 2018 *)

PARI

a(n)=(11*n^2-9*n)/2 \\ Charles R Greathouse IV, May 27 2011

Formula

a(n) = 11*n + a(n-1) - 10 with n > 0, a(0) = 0. - Vincenzo Librandi, Aug 06 2010

G.f.: x*(1+10*x)/(1-x)^3. - Bruno Berselli, Feb 04 2011

a(11*a(n) + 56*n + 1) = a(11*a(n) + 56*n) + a(11*n+1). - Vladimir Shevelev, Jan 24 2014

Product_{n>=2} (1 - 1/a(n)) = 11/13. - Amiram Eldar, Jan 21 2021

E.g.f.: exp(x)*(x + 11*x^2/2). - Nikolaos Pantelidis, Feb 06 2023