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URL: https://oeis.org/A069133

⇱ A069133 - OEIS

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A069133
Centered 20-gonal (or icosagonal) numbers.
15
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
1,2
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
REFERENCES
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 51.
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
KEYWORD
easy,nice,nonn
AUTHOR
Terrel Trotter, Jr., Apr 07 2002
STATUS
approved