A051867 15-gonal (or pentadecagonal) numbers: n*(13n-11)/2.

0, 1, 15, 42, 82, 135, 201, 280, 372, 477, 595, 726, 870, 1027, 1197, 1380, 1576, 1785, 2007, 2242, 2490, 2751, 3025, 3312, 3612, 3925, 4251, 4590, 4942, 5307, 5685, 6076, 6480, 6897, 7327, 7770, 8226, 8695, 9177, 9672, 10180, 10701

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, 15,... and the parallel line from 1, in the direction 1, 42,..., in the square spiral whose vertices are the generalized 15-gonal numbers. - 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

• Harvey P. Dale, 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

Table[n (13n-11)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,1,15},50] (* Harvey P. Dale, Feb 29 2012 *)

PARI

a(n)=n*(13*n-11)/2 \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: x*(1+12*x)/(1-x)^3. - Bruno Berselli, Feb 04 2011

a(n) = 13*n+a(n-1)-12 (with a(0)=0) - Vincenzo Librandi, Aug 06 2010

a(0)=0, a(1)=1, a(2)=15, a(n)=3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Feb 29 2012

a(13*a(n)+79*n+1) = a(13*a(n)+79*n) + a(13*n+1). - Vladimir Shevelev, Jan 24 2014

Product_{n>=2} (1 - 1/a(n)) = 13/15. - Amiram Eldar, Jan 21 2021

E.g.f.: exp(x)*(x + 13*x^2/2). - Nikolaos Pantelidis, Feb 06 2023

a(n) = A000326(3*n-2) - 7*(n-1)^2. In general, if we let P(k,n) = the n-th k-gonal number, then P(5*k,n) = P(5,k*n-k+1) - A005449(k-1)*(n-1)^2. More generally, if we let SP(k,n) = the n-th second k-gonal number, then for m>2 and k>0, P(m*k,n) = P(m,k*n-k+1) - SP(m,k-1)*(n-1)^2. - Charlie Marion, May 21 2024