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

1,2

Comments

Centered hendecagonal (11-gonal) numbers. - Omar E. Pol, Oct 03 2011

Numbers of the form (2*m+1)^2 + k*m*(m+1)/2: in this case is k=3. See also A254963. - Bruno Berselli, Feb 11 2015

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).

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)

a(n) = A003154(n) - A000217(n-1). - Leo Tavares, Mar 29 2022

E.g.f.: exp(x)*(1 + 11*x^2/2) - 1. - Elmo R. Oliveira, Oct 18 2024

Example

a(5)=111 because 111 = (11*5^2 - 11*5 + 2)/2 = (275 - 55 + 2)/2 = 222/2.

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 *)

Prog

(PARI) a(n)=(11*n^2-11*n+2)/2 \\ Charles R Greathouse IV, Sep 24 2015

Crossrefs

Cf. A001263, A001844, A003215, A005448, A005891, A069099.

Cf. A000217, A003154, A152740, A254963.

Sequence in context: A098080 A068517 A113748 * A142245 A139635 A124705

Adjacent sequences: A069122 A069123 A069124 * A069126 A069127 A069128

Keyword

nonn,easy,nice

Author

Terrel Trotter, Jr., Apr 07 2002

Extensions

More terms from Harvey P. Dale, Jun 25 2011

Name rewritten by Bruno Berselli, Feb 11 2015

Status

approved