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A033579
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Four times pentagonal numbers: a(n) = 2*n*(3*n-1).
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24
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0, 4, 20, 48, 88, 140, 204, 280, 368, 468, 580, 704, 840, 988, 1148, 1320, 1504, 1700, 1908, 2128, 2360, 2604, 2860, 3128, 3408, 3700, 4004, 4320, 4648, 4988, 5340, 5704, 6080, 6468, 6868, 7280, 7704, 8140, 8588, 9048, 9520, 10004, 10500, 11008, 11528, 12060
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OFFSET
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0,2
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COMMENTS
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Subsequence of A062717: A010052(6*a(n)+1) = 1. - Reinhard Zumkeller, Feb 21 2011
Sequence found by reading the line from 0, in the direction 0, 4, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011
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LINKS
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Ivan Panchenko, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pentagonal Number
Wikipedia, Pentagonal number
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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a(n) = 4*n*(3*n-1)/2 = 6*n^2 - 2*n = 4*A000326(n). - Omar E. Pol, Dec 11 2008
a(n) = 2*A049450(n). - Omar E. Pol, Dec 13 2008
a(n) = a(n-1) + 12*n - 8 for n > 0, a(0)=0. - Vincenzo Librandi, Aug 05 2010
a(n) = A014642(n)/2. - Omar E. Pol, Aug 19 2011
G.f.: x*(4+8*x)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 06 2012
a(n) = A191967(2*n). - Reinhard Zumkeller, Jul 07 2012
a(n) = A181617(n+1) - A181617(n). - J. M. Bergot, Jun 28 2013
a(n) = (A174371(n) - 1)/6. - Miquel Cerda, Jul 28 2016
From Ilya Gutkovskiy, Jul 28 2016: (Start)
E.g.f.: 2*x*(2 + 3*x)*exp(x).
a(n+1) = Sum_{k=0..n} A017569(k).
Sum_{i>0} 1/a(i) = (9*log(3) - sqrt(3)*Pi)/12 = 0.3705093754425278... (End)
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MAPLE
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seq(4*binomial(3*n, 2)/3, n=0..45); # G. C. Greubel, Oct 09 2019
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MATHEMATICA
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4 PolygonalNumber[5, Range[0, 45]] (* Michael De Vlieger, Aug 02 2016, Version 10.4 *)
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PROG
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(PARI) a(n)=2*n*(3*n-1) \\ Charles R Greathouse IV, Jun 28 2013
(MAGMA) [4*Binomial(3*n, 2)/3: n in [0..45]]; // G. C. Greubel, Oct 09 2019
(Sage) [4*binomial(3*n, 2)/3 for n in (0..45)] # G. C. Greubel, Oct 09 2019
(GAP) List([0..45], n-> 4*Binomial(3*n, 2)/3 ); # G. C. Greubel, Oct 09 2019
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CROSSREFS
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Cf. A000326, A001318, A014642, A033579, A033580, A033581, A049450, A186423.
Sequence in context: A163365 A145194 A164924 * A294630 A160799 A187274
Adjacent sequences: A033576 A033577 A033578 * A033580 A033581 A033582
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Michel Marcus, Mar 04 2014
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STATUS
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approved
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