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

Author

Terrel Trotter, Jr., Apr 10 2002

Keywords

• nonn
• easy

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

Examples

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

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.
• R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
• Index entries for sequences related to centered polygonal numbers.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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.

Programs

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 *)

PARI

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

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

Extensions

More terms from Harvey P. Dale, Jul 17 2011