A069125 a(n) = (11*n^2 - 11*n + 2)/2.
1, 12, 34, 67, 111, 166, 232, 309, 397, 496, 606, 727, 859, 1002, 1156, 1321, 1497, 1684, 1882, 2091, 2311, 2542, 2784, 3037, 3301, 3576, 3862, 4159, 4467, 4786, 5116, 5457, 5809, 6172, 6546, 6931, 7327, 7734, 8152, 8581, 9021, 9472, 9934, 10407, 10891, 11386, 11892
Offset: 1
Examples
a(5)=111 because 111 = (11*5^2 - 11*5 + 2)/2 = (275 - 55 + 2)/2 = 222/2.
Links
- T. D. Noe, Table of n, a(n) for n = 1..1000
- X. Acloque, Polynexus Numbers and other mathematical wonders. [broken link]
- Leo Tavares, Illustration: Clipped Stars.
- 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
Programs
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Mathematica
FoldList[#1 + #2 &, 1, 11 Range@ 45] (* Robert G. Wilson v *) Table[(11n^2-11n+2)/2,{n,60}] (* or *) LinearRecurrence[{3,-3,1},{1,12,34},60] (* Harvey P. Dale, Jun 25 2011 *)
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PARI
a(n)=(11*n^2-11*n+2)/2 \\ Charles R Greathouse IV, Sep 24 2015
Formula
a(n) = 1 + Sum_{j=0..n-1} (11*j). - Xavier Acloque, Oct 22 2003
Binomial transform of [1, 11, 11, 0, 0, 0, ...]; Narayana transform (A001263) of [1, 11, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007
a(n) = 11*n + a(n-1) - 11 with n > 1, a(1)=1. - Vincenzo Librandi, Aug 08 2010
G.f.: -x*(1+9*x+x^2)/(x-1)^3. - R. J. Mathar, Jun 05 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=12, a(2)=34. - Harvey P. Dale, Jun 25 2011
a(n) = A152740(n-1) + 1. - Omar E. Pol, Oct 03 2011
From Amiram Eldar, Jun 21 2020: (Start)
Sum_{n>=1} 1/a(n) = 2*Pi*tan(sqrt(3/11)*Pi/2)/sqrt(33).
Sum_{n>=1} a(n)/n! = 13*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 13/(2*e) - 1. (End)
E.g.f.: exp(x)*(1 + 11*x^2/2) - 1. - Elmo R. Oliveira, Oct 18 2024
Extensions
More terms from Harvey P. Dale, Jun 25 2011
Name rewritten by Bruno Berselli, Feb 11 2015
Comments