A069133 Centered 20-gonal (or icosagonal) numbers.

1, 21, 61, 121, 201, 301, 421, 561, 721, 901, 1101, 1321, 1561, 1821, 2101, 2401, 2721, 3061, 3421, 3801, 4201, 4621, 5061, 5521, 6001, 6501, 7021, 7561, 8121, 8701, 9301, 9921, 10561, 11221, 11901, 12601, 13321, 14061, 14821, 15601, 16401, 17221, 18061, 18921, 19801

Offset: 1

Author

Terrel Trotter, Jr., Apr 07 2002

Keywords

• easy
• nice
• nonn

Comments

Equals binomial transform of [1, 20, 20, 0, 0, 0, ...]. - Gary W. Adamson, Jun 13 2008

Equals Narayana transform (A001263) of [1, 20, 0, 0, 0, ...]. - Gary W. Adamson, Jul 28 2011

Sequence found by reading the line from 1, in the direction 1, 21, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. Semi-axis opposite to A033583 in the same spiral. - Omar E. Pol, Sep 16 2011

Examples

			a(5)=201 because 201 = 10*5^2 - 10*5 + 1 = 250 - 50 + 1.
		

Links

• Ivan Panchenko, Table of n, a(n) for n = 1..1000
• John Elias, Illustration of initial terms.
• 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 polygonal numbers listed in A069190.

Programs

Magma

[10*n^2 - 10*n + 1 : n in [1..60]]; // Wesley Ivan Hurt, Oct 10 2021

Mathematica

FoldList[#1 + #2 &, 1, 20 Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
Table[10n^2-10n+1,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{1,21,61}, 50]  (* Harvey P. Dale, Apr 29 2011 *)

PARI

a(n)=10*n*(n-1)+1 \\ Charles R Greathouse IV, Jul 29 2011

Formula

a(n) = 10n^2 - 10n + 1.

a(n) = 20*n + a(n-1) - 20 with a(1)=1. - Vincenzo Librandi, Aug 08 2010

G.f.: x*(1 + 18*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)=21, a(2)=61. - Harvey P. Dale, Apr 29 2011

From Amiram Eldar, Jun 21 2020: (Start)

Sum_{n>=1} 1/a(n) = Pi*tan(sqrt(3/5)*Pi/2)/(2*sqrt(15)).

Sum_{n>=1} a(n)/n! = 11*e - 1.

Sum_{n>=1} (-1)^n * a(n)/n! = 11/e - 1. (End)

a(n) = 12*A000217(n-1) + A016754(n-1). - John Elias, Oct 23 2020

E.g.f.: exp(x)*(1 + 10*x^2) - 1. - Nikolaos Pantelidis, Feb 06 2023