A069132 Centered 19-gonal numbers.
1, 20, 58, 115, 191, 286, 400, 533, 685, 856, 1046, 1255, 1483, 1730, 1996, 2281, 2585, 2908, 3250, 3611, 3991, 4390, 4808, 5245, 5701, 6176, 6670, 7183, 7715, 8266, 8836, 9425, 10033, 10660, 11306, 11971, 12655, 13358, 14080, 14821, 15581, 16360, 17158
Offset: 1
Examples
a(5)= 191 because (19*5^2 - 19*5 + 2)/2 = (475 - 95 + 2)/2 = 382/2 = 191.
Links
- Ivan Panchenko, Table of n, a(n) for n = 1..1000
- Eric Weisstein's World of Mathematics, Centered Polygonal Numbers
- Index entries for sequences related to centered polygonal numbers
- Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Crossrefs
Cf. centered polygonal numbers listed in A069190.
Programs
-
Mathematica
FoldList[#1 + #2 &, 1, 19 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *) Table[(19n^2-19n+2)/2,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,20,58},50] (* Harvey P. Dale, Aug 21 2011 *)
-
PARI
a(n)=19*binomial(n,2)+1 \\ Charles R Greathouse IV, Jul 29 2011
Formula
a(n) = (19*n^2 - 19*n + 2)/2.
a(n) = 19*n + a(n-1) - 19 (with a(1)=1). - Vincenzo Librandi, Aug 08 2010
G.f.: x*(1 + 17*x + x^2) / (1-x)^3. - R. J. Mathar, Feb 04 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=20, a(2)=58. - Harvey P. Dale, Aug 21 2011
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = 2*Pi*tan(sqrt(11/19)*Pi/2)/sqrt(209).
Sum_{n>=1} a(n)/n! = 21*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 21/(2*e) - 1. (End)
E.g.f.: exp(x)*(1 + 19*x^2/2) - 1. - Nikolaos Pantelidis, Feb 06 2023
Comments