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.
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 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)
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
KEYWORD
easy,nice,nonn
AUTHOR
Terrel Trotter, Jr., Apr 07 2002
STATUS
approved