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%I #80 May 25 2022 10:48:48
%S 1,12,35,70,117,176,247,330,425,532,651,782,925,1080,1247,1426,1617,
%T 1820,2035,2262,2501,2752,3015,3290,3577,3876,4187,4510,4845,5192,
%U 5551,5922,6305,6700,7107,7526,7957,8400,8855,9322,9801,10292,10795,11310,11837
%N Pentagonal numbers with odd index: a(n) = (2*n+1)*(3*n+1).
%C 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
%C 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
%C 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
%C a(n) is the sum of 2*n+1 consecutive integers starting from 2*n+1. - _Bruno Berselli_, Jan 16 2018
%H Michael De Vlieger, <a href="/A033570/b033570.txt">Table of n, a(n) for n = 0..10000</a>
%H John Elias, <a href="/A033570/a033570.png">Illustration: Natural number stars</a>.
%H Leo Tavares, <a href="/A033570/a033570.jpg">Illustration: Square Block Triangles</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentagonalNumber.html">Pentagonal Number</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pentagonal_number">Pentagonal number</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: (1 + 9*x + 2*x^2)/(1-x)^3.
%F a(n) = a(n-1) + 12*n-1 for n > 0, a(0)=1. - _Vincenzo Librandi_, Nov 17 2010
%F a(n) = A000326(2*n+1) = A191967(2*n+1). - _Reinhard Zumkeller_, Jul 07 2012
%F a(n) = Sum_{i=1..2*(n+1)-1} 4*(n+1) - 2 - i. - _Wesley Ivan Hurt_, Mar 18 2014
%F E.g.f.: (1 + 11*x + 6*x^2)*exp(x). - _G. C. Greubel_, Oct 12 2019
%F From _Amiram Eldar_, Feb 20 2022: (Start)
%F Sum_{n>=0} 1/a(n) = Pi/(2*sqrt(3)) - 2*log(2) + 3*log(3)/2.
%F Sum_{n>=0} (-1)^n/a(n) = (1/sqrt(3) - 1/2)*Pi + log(2). (End)
%F a(n) = A016754(n) + A014105(n). - _Leo Tavares_, May 24 2022
%p A033570:=n->(2*n+1)*(3*n+1); seq(A033570(n), n=0..40); # _Wesley Ivan Hurt_, Mar 18 2014
%t LinearRecurrence[{3,-3,1},{1,12,35},50]
%t Table[(2 n + 1) (3 n + 1), {n, 0, 50}] (* or *)
%t CoefficientList[Series[(1 + 9 x + 2 x^2)/(1 - x)^3, {x, 0, 50}], x] (* _Michael De Vlieger_, Dec 12 2016 *)
%t PolygonalNumber[5,Range[1,101,2]] (* _Harvey P. Dale_, Aug 02 2021 *)
%o (PARI) a(n)=(2*n+1)*(3*n+1) \\ _Charles R Greathouse IV_, Jun 11 2015
%o (Magma) [(2*n+1)*(3*n+1) : n in [0..50]]; // _Wesley Ivan Hurt_, Dec 11 2016
%o (Sage) [(2*n+1)*(3*n+1) for n in range(50)] # _G. C. Greubel_, Oct 12 2019
%o (GAP) List([0..50], n-> (2*n+1)*(3*n+1)); # _G. C. Greubel_, Oct 12 2019
%Y Cf. A000326, A001318, A033568, A049452, A049453, A191967.
%Y Cf. A016754, A014105.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Ray Chandler_, Dec 08 2011
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