A051873 21-gonal numbers: a(n) = n*(19n - 17)/2.

0, 1, 21, 60, 118, 195, 291, 406, 540, 693, 865, 1056, 1266, 1495, 1743, 2010, 2296, 2601, 2925, 3268, 3630, 4011, 4411, 4830, 5268, 5725, 6201, 6696, 7210, 7743, 8295, 8866, 9456, 10065, 10693, 11340, 12006, 12691, 13395, 14118

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, 21, ... and the parallel line from 1, in the direction 1, 60, ..., in the square spiral whose vertices are the generalized 21-gonal numbers. - Omar E. Pol, Jul 18 2012

Partial sums of A215144. - Leo Tavares, Mar 17 2023

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

• Vincenzo Librandi, 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. A215144, A226490.

Programs

Maple

A051873 := proc(n) n*(19*n-17)/2 ;end proc: seq(A051873(n),n=0..30) ; # R. J. Mathar, Feb 05 2011

Mathematica

PolygonalNumber[21,Range[0,40]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 22 2016 *)
Table[n*(19*n - 17)/2, {n, 0, 100}] (* Robert Price, Oct 11 2018 *)

PARI

n*(19*n-17)/2 \\ Charles R Greathouse IV, Jan 24 2014

Formula

G.f.: x*(1+18*x)/(1-x)^3. - Bruno Berselli, Feb 04 2011

a(n) = 19*n+a(n-1)-18 with n>0, a(0)=0. - Vincenzo Librandi, Aug 06 2010

a(n) = A226490(n) - n. - Bruno Berselli, Jun 11 2013

a(19*a(n)+172*n+1) = a(19*a(n)+172*n) + a(19*n+1). - Vladimir Shevelev, Jan 24 2014

Product_{n>=2} (1 - 1/a(n)) = 19/21. - Amiram Eldar, Jan 22 2021

E.g.f.: exp(x)*(x + 19*x^2/2). - Nikolaos Pantelidis, Feb 06 2023