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

Offset

1,2

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

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

Index entries for sequences related to centered polygonal numbers

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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

Example

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

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 *)

Prog

(PARI) a(n)=10*n*(n-1)+1 \\ Charles R Greathouse IV, Jul 29 2011
(Magma) [10*n^2 - 10*n + 1 : n in [1..60]]; // Wesley Ivan Hurt, Oct 10 2021

Crossrefs

Cf. centered polygonal numbers listed in A069190.

Sequence in context: A223460 A219690 A325319 * A124714 A126375 A146468

Adjacent sequences: A069130 A069131 A069132 * A069134 A069135 A069136

Keyword

easy,nice,nonn

Author

Terrel Trotter, Jr., Apr 07 2002

Status

approved