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A069133
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Centered 20-gonal (or icosagonal) numbers.
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13
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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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).
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FORMULA
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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
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EXAMPLE
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a(5)=201 because 201 = 10*5^2 - 10*5 + 1 = 250 - 50 + 1.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. centered polygonal numbers listed in A069190.
Sequence in context: A223460 A219690 A325319 * A124714 A126375 A146468
Adjacent sequences: A069130 A069131 A069132 * A069134 A069135 A069136
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KEYWORD
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easy,nice,nonn
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AUTHOR
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Terrel Trotter, Jr., Apr 07 2002
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STATUS
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approved
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