A049452 Pentagonal numbers with even index.

0, 5, 22, 51, 92, 145, 210, 287, 376, 477, 590, 715, 852, 1001, 1162, 1335, 1520, 1717, 1926, 2147, 2380, 2625, 2882, 3151, 3432, 3725, 4030, 4347, 4676, 5017, 5370, 5735, 6112, 6501, 6902, 7315, 7740, 8177, 8626, 9087, 9560, 10045, 10542

Offset: 0

Author

Joe Keane (jgk(AT)jgk.org)

Keywords

• nonn
• easy

Comments

If Y is a 3-subset of an (2n+1)-set X then, for n>=4, a(n-1) 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 0, in the direction 0, 5,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011

a(n) is the sum of 2*n consecutive integers starting from 2*n. - Bruno Berselli, Jan 16 2018

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000326, A033570, A049453, A001318, A033568, A185019.

See index to sequences with numbers of the form n*(d*n+10-d)/2 in A140090.

Programs

Maple

seq(n*(6*n-1),n=0..42); # Zerinvary Lajos, Jun 12 2007

Mathematica

Table[n(6n-1),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,5,22},50] (* Harvey P. Dale, Mar 07 2012 *)

PARI

a(n)=n*(6*n-1) \\ Charles R Greathouse IV, Sep 24 2015

Formula

a(n) = n*(6*n-1).

G.f.: x*(5+7*x)/(1-x)^3.

a(n) = C(6*n,2)/3. - Zerinvary Lajos, Jan 02 2007

a(n) = A001105(n) + A033991(n) = A033428(n) + A049450(n) = A022266(n) + A000326(n). - Zerinvary Lajos, Jun 12 2007

a(n) = 12*n + a(n-1) - 7 for n>0, a(0)=0. - Vincenzo Librandi, Aug 06 2010

a(n) = 4*A000217(n) + A001107(n). - Bruno Berselli, Feb 11 2011

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=5, a(2)=22. - Harvey P. Dale, Mar 07 2012

E.g.f.: (6*x^2 + 5*x)*exp(x). - G. C. Greubel, Jul 17 2017

From Amiram Eldar, Jul 03 2020: (Start)

Sum_{n>=1} 1/a(n) = 2*log(2) + 3*log(3)/2 - sqrt(3)*Pi/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi - log(2) - 2*sqrt(3)*arccoth(sqrt(3)). (End)