OFFSET
1,2
COMMENTS
Binomial transform of [1, 19, 19, 0, 0, 0, ...] and Narayana transform (A001263) of [1, 19, 0, 0, 0, ...]. - Gary W. Adamson, Jul 28 2011
LINKS
Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Centered Polygonal Numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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
EXAMPLE
a(5)= 191 because (19*5^2 - 19*5 + 2)/2 = (475 - 95 + 2)/2 = 382/2 = 191.
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 *)
PROG
(PARI) a(n)=19*binomial(n, 2)+1 \\ Charles R Greathouse IV, Jul 29 2011
CROSSREFS
KEYWORD
easy,nice,nonn
AUTHOR
Terrel Trotter, Jr., Apr 07 2002
STATUS
approved