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A033570
Pentagonal numbers with odd index: a(n) = (2*n+1)*(3*n+1).
20
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
OFFSET
0,2
COMMENTS
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
FORMULA
G.f.: (1 + 9*x + 2*x^2)/(1-x)^3.
a(n) = a(n-1) + 12*n-1 for n > 0, a(0)=1. - Vincenzo Librandi, Nov 17 2010
a(n) = A000326(2*n+1) = A191967(2*n+1). - Reinhard Zumkeller, Jul 07 2012
a(n) = Sum_{i=1..2*(n+1)-1} 4*(n+1) - 2 - i. - Wesley Ivan Hurt, Mar 18 2014
E.g.f.: (1 + 11*x + 6*x^2)*exp(x). - G. C. Greubel, Oct 12 2019
From Amiram Eldar, Feb 20 2022: (Start)
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)
a(n) = A016754(n) + A014105(n). - Leo Tavares, May 24 2022
MAPLE
A033570:=n->(2*n+1)*(3*n+1); seq(A033570(n), n=0..40); # Wesley Ivan Hurt, Mar 18 2014
MATHEMATICA
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 *)
PROG
(PARI) a(n)=(2*n+1)*(3*n+1) \\ Charles R Greathouse IV, Jun 11 2015
(Magma) [(2*n+1)*(3*n+1) : n in [0..50]]; // Wesley Ivan Hurt, Dec 11 2016
(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
KEYWORD
nonn,easy
EXTENSIONS
More terms from Ray Chandler, Dec 08 2011
STATUS
approved