A069190 Centered 24-gonal numbers.

1, 25, 73, 145, 241, 361, 505, 673, 865, 1081, 1321, 1585, 1873, 2185, 2521, 2881, 3265, 3673, 4105, 4561, 5041, 5545, 6073, 6625, 7201, 7801, 8425, 9073, 9745, 10441, 11161, 11905, 12673, 13465, 14281, 15121, 15985, 16873, 17785, 18721, 19681, 20665, 21673

Offset

1,2

Comments

Sequence found by reading the line from 1, in the direction 1, 25, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Semi-axis opposite to A135453 in the same spiral. - Omar E. Pol, Sep 16 2011

Links

Ivan Panchenko, Table of n, a(n) for n = 1..1000

John Elias, Illustration: Odd Ordered Star Perimeters.

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) = 12*n^2 - 12*n + 1.

a(n) = 24*n + a(n-1) - 24 with a(1)=1. - Vincenzo Librandi, Aug 08 2010

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=1, a(2)=25, a(3)=73. - Harvey P. Dale, Jul 17 2011

G.f.: x*(1+22*x+x^2)/(1-x)^3. - Harvey P. Dale, Jul 17 2011

Binomial transform of [1, 24, 24, 0, 0, 0, ...] and Narayana transform (cf. A001263) of [1, 24, 0, 0, 0, ...]. - Gary W. Adamson, Jul 26 2011

From Amiram Eldar, Jun 21 2020: (Start)

Sum_{n>=1} 1/a(n) = Pi*tan(Pi/sqrt(6))/(4*sqrt(6)).

Sum_{n>=1} a(n)/n! = 13*e - 1.

Sum_{n>=1} (-1)^n * a(n)/n! = 13/e - 1. (End)

E.g.f.: exp(x)*(1 + 12*x^2) - 1. - Stefano Spezia, May 31 2022

Example

a(5) = 241 because 12*5^2 - 12*5 + 1 = 300 - 60 + 1 = 241.

Mathematica

FoldList[#1 + #2 &, 1, 24 Range@ 45] (* Robert G. Wilson v *)
Table[12n^2-12n+1, {n, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 25, 73}, 50] (* Harvey P. Dale, Jul 17 2011 *)

Prog

(PARI) a(n)=12*n^2-12*n+1 \\ Charles R Greathouse IV, Sep 24 2015

Crossrefs

Cf. centered k-gonal numbers with k=3..25: A005448, A001844, A005891, A003215, A069099, A016754, A060544, A062786, A069125, A003154, A069126, A069127, A069128, A069129, A069130, A069131, A069132, A069133, A069178, A069173, A069174, A069190, A262221.

Sequence in context: A255184 A235941 A337156 * A124718 A126379 A114553

Adjacent sequences: A069187 A069188 A069189 * A069191 A069192 A069193

Keyword

nonn,easy

Author

Terrel Trotter, Jr., Apr 10 2002

Extensions

More terms from Harvey P. Dale, Jul 17 2011

Status

approved