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A033570 Pentagonal numbers with odd index: a(n) = (2*n+1)*(3*n+1).

%I A033570 #81 Feb 16 2025 08:32:36
%S A033570 1,12,35,70,117,176,247,330,425,532,651,782,925,1080,1247,1426,1617,
%T A033570 1820,2035,2262,2501,2752,3015,3290,3577,3876,4187,4510,4845,5192,
%U A033570 5551,5922,6305,6700,7107,7526,7957,8400,8855,9322,9801,10292,10795,11310,11837
%N A033570 Pentagonal numbers with odd index: a(n) = (2*n+1)*(3*n+1).
%C A033570 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 A033570 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 A033570 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 A033570 a(n) is the sum of 2*n+1 consecutive integers starting from 2*n+1. - _Bruno Berselli_, Jan 16 2018
%H A033570 Michael De Vlieger, <a href="/A033570/b033570.txt">Table of n, a(n) for n = 0..10000</a>
%H A033570 John Elias, <a href="/A033570/a033570.png">Illustration: Natural number stars</a>.
%H A033570 Leo Tavares, <a href="/A033570/a033570.jpg">Illustration: Square Block Triangles</a>
%H A033570 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalNumber.html">Pentagonal Number</a>.
%H A033570 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pentagonal_number">Pentagonal number</a>.
%H A033570 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A033570 G.f.: (1 + 9*x + 2*x^2)/(1-x)^3.
%F A033570 a(n) = a(n-1) + 12*n-1 for n > 0, a(0)=1. - _Vincenzo Librandi_, Nov 17 2010
%F A033570 a(n) = A000326(2*n+1) = A191967(2*n+1). - _Reinhard Zumkeller_, Jul 07 2012
%F A033570 a(n) = Sum_{i=1..2*(n+1)-1} 4*(n+1) - 2 - i. - _Wesley Ivan Hurt_, Mar 18 2014
%F A033570 E.g.f.: (1 + 11*x + 6*x^2)*exp(x). - _G. C. Greubel_, Oct 12 2019
%F A033570 From _Amiram Eldar_, Feb 20 2022: (Start)
%F A033570 Sum_{n>=0} 1/a(n) = Pi/(2*sqrt(3)) - 2*log(2) + 3*log(3)/2.
%F A033570 Sum_{n>=0} (-1)^n/a(n) = (1/sqrt(3) - 1/2)*Pi + log(2). (End)
%F A033570 a(n) = A016754(n) + A014105(n). - _Leo Tavares_, May 24 2022
%p A033570 A033570:=n->(2*n+1)*(3*n+1); seq(A033570(n), n=0..40); # _Wesley Ivan Hurt_, Mar 18 2014
%t A033570 LinearRecurrence[{3,-3,1},{1,12,35},50]
%t A033570 Table[(2 n + 1) (3 n + 1), {n, 0, 50}] (* or *)
%t A033570 CoefficientList[Series[(1 + 9 x + 2 x^2)/(1 - x)^3, {x, 0, 50}], x] (* _Michael De Vlieger_, Dec 12 2016 *)
%t A033570 PolygonalNumber[5,Range[1,101,2]] (* _Harvey P. Dale_, Aug 02 2021 *)
%o A033570 (PARI) a(n)=(2*n+1)*(3*n+1) \\ _Charles R Greathouse IV_, Jun 11 2015
%o A033570 (Magma) [(2*n+1)*(3*n+1) : n in [0..50]]; // _Wesley Ivan Hurt_, Dec 11 2016
%o A033570 (Sage) [(2*n+1)*(3*n+1) for n in range(50)] # _G. C. Greubel_, Oct 12 2019
%o A033570 (GAP) List([0..50], n-> (2*n+1)*(3*n+1)); # _G. C. Greubel_, Oct 12 2019
%Y A033570 Cf. A000326, A001318, A033568, A049452, A049453, A191967.
%Y A033570 Cf. A016754, A014105.
%K A033570 nonn,easy
%O A033570 0,2
%A A033570 _N. J. A. Sloane_
%E A033570 More terms from _Ray Chandler_, Dec 08 2011