A051866 14-gonal (or tetradecagonal) numbers: a(n) = n*(6*n-5).

0, 1, 14, 39, 76, 125, 186, 259, 344, 441, 550, 671, 804, 949, 1106, 1275, 1456, 1649, 1854, 2071, 2300, 2541, 2794, 3059, 3336, 3625, 3926, 4239, 4564, 4901, 5250, 5611, 5984, 6369, 6766, 7175, 7596, 8029, 8474, 8931, 9400, 9881, 10374

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, 14, ... and the parallel line from 1, in the direction 1, 39, ..., in the square spiral whose vertices are the generalized 14-gonal numbers A195818. Also sequence found by reading the segment (0, 1) together with the line from 1, in the direction 1, 14, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Jul 18 2012

After 0, partial sums of A017533. - Bruno Berselli, Sep 11 2013

This is also a star heptagonal number: a(n) = A000566(n) + 7*A000217(n-1). - Luciano Ancora, Mar 30 2015

Starting with offset 1, this is the binomial transform of (1, 13, 12, 0, 0, 0, ...). - Gary W. Adamson, Jul 29 2015

References

• Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 189.
• Elena Deza and Michel Marie 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).

Crossrefs

Cf. A000217, A000566, A001318, A017533, A033568, A195818.

Programs

Maple

A051866 := proc(n) n*(6*n-5) ; end proc: seq(A051866(n),n=0..30) ; # R. J. Mathar, Feb 05 2011

Mathematica

Table[n*(6*n - 5), {n, 0, 100}] (* Robert Price, Oct 11 2018 *)

PARI

a(n)=n*(6*n-5); \\ Joerg Arndt, Feb 01 2014

Formula

G.f.: x*(1+11*x)/(1-x)^3. - Bruno Berselli, Feb 04 2011

a(n) = 12*n + a(n-1) - 11, with n > 0, a(0)=0. - Vincenzo Librandi, Aug 06 2010

a(n) = A033568(n) - 1. - Omar E. Pol, Jul 18 2012

a(12*a(n)+67*n+1) = a(12*a(n) + 67*n) + a(12*n + 1). - Vladimir Shevelev, Jan 24 2014

From Amiram Eldar, Oct 20 2020: (Start)

Sum_{n>=1} 1/a(n) = (sqrt(3)*Pi + log(432))/10.

Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi + 2*sqrt(3)*arccoth(sqrt(3)) - log(2))/5. (End)

Product_{n>=2} (1 - 1/a(n)) = 6/7. - Amiram Eldar, Jan 21 2021

E.g.f.: exp(x)*x*(1 + 6*x). - Stefano Spezia, Jun 08 2021