A051875 23-gonal numbers: a(n) = n(21n-19)/2.

0, 1, 23, 66, 130, 215, 321, 448, 596, 765, 955, 1166, 1398, 1651, 1925, 2220, 2536, 2873, 3231, 3610, 4010, 4431, 4873, 5336, 5820, 6325, 6851, 7398, 7966, 8555, 9165, 9796, 10448, 11121, 11815, 12530, 13266, 14023, 14801, 15600

Offset

0,3

Comments

Sequence found by reading the line from 0, in the direction 0, 23, ..., and the parallel line from 1, in the direction 1, 66, ..., in the square spiral whose vertices are the generalized 23-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

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).

Formula

G.f.: x*(1+20*x)/(1-x)^3. - Bruno Berselli, Feb 04 2011

a(n) = 21*n + a(n-1) - 20 with n > 0, a(0) = 0. - Vincenzo Librandi, Aug 06 2010

a(n) = A226491(n) - n. - Bruno Berselli, Jun 11 2013

a(21*a(n)+211*n+1) = a(21*a(n)+211*n) + a(21*n+1). - Vladimir Shevelev, Jan 24 2014

Product_{n>=2} (1 - 1/a(n)) = 21/23. - Amiram Eldar, Jan 22 2021

E.g.f.: exp(x)*(x + 21*x^2/2). - Nikolaos Pantelidis, Feb 06 2023

Mathematica

CoefficientList[Series[x (1 + 20 x) / (1 - x)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 19 2013 *)
Table[(21n^2 - 19n)/2, {n, 0, 39}] (* Alonso del Arte, Jan 23 2015 *)
PolygonalNumber[23, Range[0, 40]] (* Harvey P. Dale, Aug 01 2022 *)

Prog

(PARI) a(n)=n*(21*n-19)/2 \\ Charles R Greathouse IV, Jan 24 2014

Crossrefs

Sequence in context: A316578 A323220 A001346 * A125872 A228611 A104945

Adjacent sequences: A051872 A051873 A051874 * A051876 A051877 A051878

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 15 1999

Status

approved