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A135705
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a(n) = 10*binomial(n,2) + 9*n.
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13
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0, 9, 28, 57, 96, 145, 204, 273, 352, 441, 540, 649, 768, 897, 1036, 1185, 1344, 1513, 1692, 1881, 2080, 2289, 2508, 2737, 2976, 3225, 3484, 3753, 4032, 4321, 4620, 4929, 5248, 5577, 5916, 6265, 6624, 6993, 7372, 7761, 8160, 8569, 8988, 9417, 9856, 10305, 10764
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OFFSET
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0,2
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COMMENTS
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Also, second 12-gonal (or dodecagonal) numbers. Identity for the numbers b(n)=n*(h*n+h-2)/2 (see Crossrefs): sum_{i=0..n} (b(n)+i)^2 = (sum_{i=n+1..2*n} (b(n)+i)^2) + h*(h-4)*A000217(n)^2 for n>0. - Bruno Berselli, Jan 15 2011
Sequence found by reading the line from 0, in the direction 0, 28, ..., and the line from 9, in the direction 9, 57, ..., in the square spiral whose vertices are the generalized 12-gonal numbers A195162. - Omar E. Pol, Jul 24 2012
Bisection of A195162. - Omar E. Pol, Aug 04 2012
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LINKS
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Ivan Panchenko, Table of n, a(n) for n = 0..1000
L. Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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From R. J. Mathar, Mar 06 2008: (Start)
O.g.f.: x*(9+x)/(1-x)^3.
a(n) = n*(5*n+4). (End)
a(n) = a(n-1) + 10*n - 1 (with a(0)=0). - Vincenzo Librandi, Nov 24 2009
a(n) = Sum_{i=0..n-1} A017377(i) for n>0. - Bruno Berselli, Jan 15 2011
a(n) = A131242(10n+8). - Philippe Deléham, Mar 27 2013
Sum_{n>=1} 1/a(n) = 5/16 + sqrt(1 + 2/sqrt(5))*Pi/8 - 5*log(5)/16 - sqrt(5)*log((1 + sqrt(5))/2)/8 = 0.2155517745488486003038... . - Vaclav Kotesovec, Apr 27 2016
From G. C. Greubel, Oct 29 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: x*(9 + 5*x)*exp(x). (End)
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {0, 9, 28}, 50] (* or *) Table[5*n^2 + 4*n, {n, 0, 50}] (* G. C. Greubel, Oct 29 2016 *)
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PROG
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(PARI) a(n) = 10*binomial(n, 2) + 9*n \\ Charles R Greathouse IV, Jun 11 2015
(MAGMA) [n*(5*n+4): n in [0..50]]; // G. C. Greubel, Jul 04 2019
(Sage) [n*(5*n+4) for n in (0..50)] # G. C. Greubel, Jul 04 2019
(GAP) List([0..50], n-> n*(5*n+4)) # G. C. Greubel, Jul 04 2019
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CROSSREFS
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Second n-gonal numbers: A005449, A014105, A147875, A045944, A179986, A033954, A062728, this sequence.
Sequence in context: A015245 A031308 A063155 * A321559 A041359 A034126
Adjacent sequences: A135702 A135703 A135704 * A135706 A135707 A135708
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, Mar 04 2008
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STATUS
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approved
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