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A033570
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Pentagonal numbers with odd index: a(n) = (2*n+1)*(3*n+1).
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20
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1, 12, 35, 70, 117, 176, 247, 330, 425, 532, 651, 782, 925, 1080, 1247, 1426, 1617, 1820, 2035, 2262, 2501, 2752, 3015, 3290, 3577, 3876, 4187, 4510, 4845, 5192, 5551, 5922, 6305, 6700, 7107, 7526, 7957, 8400, 8855, 9322, 9801, 10292, 10795, 11310, 11837
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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 2*n-set X then, for n >= 4, a(n-2) 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 1, in the direction 1, 12, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011
If two independent real random variables, x and y, are distributed according to the same exponential distribution: pdf(x) = lambda * exp(-lambda * x), lambda > 0, then the probability that 2 <= x/(n*y) < 3 is given by n/a(n) (for n>1). - Andres Cicuttin, Dec 11 2016
a(n) is the sum of 2*n+1 consecutive integers starting from 2*n+1. - Bruno Berselli, Jan 16 2018
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LINKS
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FORMULA
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G.f.: (1 + 9*x + 2*x^2)/(1-x)^3.
Sum_{n>=0} 1/a(n) = Pi/(2*sqrt(3)) - 2*log(2) + 3*log(3)/2.
Sum_{n>=0} (-1)^n/a(n) = (1/sqrt(3) - 1/2)*Pi + log(2). (End)
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MAPLE
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {1, 12, 35}, 50]
Table[(2 n + 1) (3 n + 1), {n, 0, 50}] (* or *)
CoefficientList[Series[(1 + 9 x + 2 x^2)/(1 - x)^3, {x, 0, 50}], x] (* Michael De Vlieger, Dec 12 2016 *)
PolygonalNumber[5, Range[1, 101, 2]] (* Harvey P. Dale, Aug 02 2021 *)
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PROG
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(Sage) [(2*n+1)*(3*n+1) for n in range(50)] # G. C. Greubel, Oct 12 2019
(GAP) List([0..50], n-> (2*n+1)*(3*n+1)); # G. C. Greubel, Oct 12 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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