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A049452
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Pentagonal numbers with even index.
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32
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0, 5, 22, 51, 92, 145, 210, 287, 376, 477, 590, 715, 852, 1001, 1162, 1335, 1520, 1717, 1926, 2147, 2380, 2625, 2882, 3151, 3432, 3725, 4030, 4347, 4676, 5017, 5370, 5735, 6112, 6501, 6902, 7315, 7740, 8177, 8626, 9087, 9560, 10045, 10542
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OFFSET
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0,2
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COMMENTS
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If Y is a 3-subset of an (2n+1)-set X then, for n>=4, a(n-1) is the number of 4-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 16 2007
Sequence found by reading the line (one of the diagonal axes) from 0, in the direction 0, 5,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011
a(n) is the sum of 2*n consecutive integers starting from 2*n. - Bruno Berselli, Jan 16 2018
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LINKS
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Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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a(n) = n*(6*n-1).
G.f.: x*(5+7*x)/(1-x)^3.
a(n) = C(6*n,2)/3. - Zerinvary Lajos, Jan 02 2007
a(n) = A001105(n) + A033991(n) = A033428(n) + A049450(n) = A022266(n) + A000326(n). - Zerinvary Lajos, Jun 12 2007
a(n) = 12*n + a(n-1) - 7 for n>0, a(0)=0. - Vincenzo Librandi, Aug 06 2010
a(n) = 4*A000217(n) + A001107(n). - Bruno Berselli, Feb 11 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=5, a(2)=22. - Harvey P. Dale, Mar 07 2012
E.g.f.: (6*x^2 + 5*x)*exp(x). - G. C. Greubel, Jul 17 2017
From Amiram Eldar, Jul 03 2020: (Start)
Sum_{n>=1} 1/a(n) = 2*log(2) + 3*log(3)/2 - sqrt(3)*Pi/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi - log(2) - 2*sqrt(3)*arccoth(sqrt(3)). (End)
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MAPLE
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seq(n*(6*n-1), n=0..42); # Zerinvary Lajos, Jun 12 2007
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MATHEMATICA
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Table[n(6n-1), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 5, 22}, 50] (* Harvey P. Dale, Mar 07 2012 *)
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PROG
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(PARI) a(n)=n*(6*n-1) \\ Charles R Greathouse IV, Sep 24 2015
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CROSSREFS
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Cf. A000326, A033570, A049453, A001318, A033568, A185019.
See index to sequences with numbers of the form n*(d*n+10-d)/2 in A140090.
Sequence in context: A273648 A272993 A273581 * A273075 A272824 A273677
Adjacent sequences: A049449 A049450 A049451 * A049453 A049454 A049455
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KEYWORD
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nonn,easy
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AUTHOR
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Joe Keane (jgk(AT)jgk.org)
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STATUS
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approved
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