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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• John Elias, Illustration: Natural number stars.
• Leo Tavares, Illustration: Square Block Triangles
• Eric Weisstein's World of Mathematics, Pentagonal Number.
• Wikipedia, Pentagonal number.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000326, A001318, A033568, A049452, A049453, A191967.

Cf. A016754, A014105.

Programs

GAP

List([0..50], n-> (2*n+1)*(3*n+1)); # G. C. Greubel, Oct 12 2019

Magma

[(2*n+1)*(3*n+1) : n in [0..50]]; // Wesley Ivan Hurt, Dec 11 2016

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 *)

PARI

a(n)=(2*n+1)*(3*n+1) \\ Charles R Greathouse IV, Jun 11 2015

Sage

[(2*n+1)*(3*n+1) for n in range(50)] # G. C. Greubel, Oct 12 2019

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

Extensions

More terms from Ray Chandler, Dec 08 2011