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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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FORMULA
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a(n) = n*(6*n-1).
G.f.: x*(5+7*x)/(1-x)^3.
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
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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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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CROSSREFS
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See index to sequences with numbers of the form n*(d*n+10-d)/2 in A140090.
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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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