A051871 19-gonal (or enneadecagonal) numbers: n(17n-15)/2.

0, 1, 19, 54, 106, 175, 261, 364, 484, 621, 775, 946, 1134, 1339, 1561, 1800, 2056, 2329, 2619, 2926, 3250, 3591, 3949, 4324, 4716, 5125, 5551, 5994, 6454, 6931, 7425, 7936, 8464, 9009, 9571, 10150, 10746, 11359, 11989, 12636, 13300

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, 19, ... and the parallel line from 1, in the direction 1, 54, ..., in the square spiral whose vertices are the generalized 19-gonal numbers. - Omar E. Pol, Jul 18 2012

Partial sums of A215137 (17n + 1). - Jeremy Gardiner, Aug 04 2012

Examples

			a(1) = 17 * 1 + 0 - 16 = 1.
a(2) = 17 * 2 + 1 - 16 = 19.
a(3) = 17 * 3 + 19 - 16 = 54. - _Vincenzo Librandi_, Aug 06 2010
		

References

• Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 189.
• Elena Deza and Michel M. Deza, Figurate numbers, World Scientific Publishing (2012), page 6.

Links

• Jeremy Gardiner, Table of n, a(n) for n = 0..999
• Index to sequences related to polygonal numbers
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Programs

Maple

A051871 := proc(n) n*(17*n-15)/2 ; end proc: seq(A051871(n),n=0..30) ; # R. J. Mathar, Feb 05 2011

Mathematica

Table[(17n^2 - 15n)/2, {n, 0, 39}] (* Alonso del Arte, Feb 19 2015 *)

PARI

a(n)=n*(17*n-15)/2; \\ Charles R Greathouse IV, Jan 24 2014

Formula

a(n) = n(17n-15)/2.

G.f.: x*(1+16*x)/(1-x)^3. - Bruno Berselli, Feb 04 2011

a(n) = 17*n + a(n-1) - 16 (with a(0) = 0). - Vincenzo Librandi, Aug 06 2010

a(17*a(n) + 137*n + 1) = a(17*a(n) + 137*n) + a(17*n+1). - Vladimir Shevelev, Jan 24 2014

Product_{n>=2} (1 - 1/a(n)) = 17/19. - Amiram Eldar, Jan 22 2021

E.g.f.: exp(x)*(x + 17*x^2/2). - Nikolaos Pantelidis, Feb 06 2023