A051872 20-gonal (or icosagonal) numbers: a(n) = n*(9*n-8).

0, 1, 20, 57, 112, 185, 276, 385, 512, 657, 820, 1001, 1200, 1417, 1652, 1905, 2176, 2465, 2772, 3097, 3440, 3801, 4180, 4577, 4992, 5425, 5876, 6345, 6832, 7337, 7860, 8401, 8960, 9537, 10132, 10745, 11376, 12025, 12692, 13377, 14080

Offset: 0

Author

N. J. A. Sloane, Dec 15 1999

Keywords

• nonn
• easy

Comments

This sequence does not contain any squares other than 0 and 1. See A188896. - T. D. Noe, Apr 13 2011

Sequence found by reading the line from 0, in the direction 0, 20,... and the parallel line from 1, in the direction 1, 57,..., in the square spiral whose vertices are the generalized 20-gonal numbers. - Omar E. Pol, Jul 18 2012

This is also a star decagonal number: a(n) = A001107(n) + 10*A000217(n-1). - Luciano Ancora, Mar 30 2015

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

• Ivan Panchenko, 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

Maple

A051872 := proc(n) n*(9*n-8) ;end proc: seq(A051872(n),n=0..30) ; # R. J. Mathar, Feb 05 2011

Mathematica

Table[9n^2 - 8n, {n, 0, 59}] (* Alonso del Arte, Dec 20 2014 *)
PolygonalNumber[20,Range[0,50]] (* Requires Mathematica version 10 or later *)  (* Harvey P. Dale, May 14 2017 *)

PARI

n*(9*n-8) \\ Charles R Greathouse IV, Jan 24 2014

Formula

a(n) = 18*n + a(n-1) - 17, with n > 0, a(0) = 0. - Vincenzo Librandi, Aug 06 2010

G.f.: x*(1+17*x)/(1-x)^3. - Bruno Berselli, Feb 04 2011

a(18*a(n) + 154*n + 1) = a(18*a(n) + 154*n) + a(18*n + 1). - Vladimir Shevelev, Jan 24 2014

Product_{n>=2} (1 - 1/a(n)) = 9/10. - Amiram Eldar, Jan 22 2021

For n>0, a(n) = A002378(3*n-2) + n - 2. - Charlie Marion, Jul 18 2022

E.g.f.: exp(x)*(x + 9*x^2). - Nikolaos Pantelidis, Feb 05 2023