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 A069133 Centered 20-gonal (or icosagonal) numbers. 14
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified September 28 02:54 EDT 2023. Contains 365714 sequences. (Running on oeis4.)