A045991 a(n) = n^3 - n^2.

0, 0, 4, 18, 48, 100, 180, 294, 448, 648, 900, 1210, 1584, 2028, 2548, 3150, 3840, 4624, 5508, 6498, 7600, 8820, 10164, 11638, 13248, 15000, 16900, 18954, 21168, 23548, 26100, 28830, 31744, 34848, 38148, 41650, 45360, 49284, 53428, 57798, 62400, 67240, 72324

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of edges in the line graph of the complete bipartite graph of order 2n, L(K_n,n). - Roberto E. Martinez II, Jan 07 2002

Number of edges of the Cartesian product of two complete graphs K_n X K_n. - Roberto E. Martinez II, Jan 07 2002

That is, number of edges in the n X n rook graph. - Eric W. Weisstein, Jun 20 2017

n such that x^3 + x^2 + n factors over the integers. - James R. Buddenhagen, Apr 19 2005

Also the number of triangles in a 2 X n grid of points and therefore also (n choose 2) * (n choose 1) * 2, or (2n choose 3) - 2*(n choose 3). - Joshua Zucker, Jan 11 2006

Nonnegative X values of solutions to the equation (X-Y)^3-XY=0. To find Y values: b(n)=(n+1)*n^2 (see A011379). I proved that, if(X,Y) is different from (0,0) and m=2, 4, 6, 8, 10, 12,..., then the equation (X-Y)^m-XY=0,... has no solution. - Mohamed Bouhamida, May 10 2006

For n>=1, a(n) is equal to the number of functions f:{1,2,3}->{1,2,...,n} such that for a fixed x in {1,2,3} and a fixed y in {1,2,...,n} we have f(x)<>y. - Aleksandar M. Janjic and Milan Janjic, Mar 13 2007

a(n) equals the coefficient of log(2) in 2F1(n-1,n-1,n+1,-1). - John M. Campbell, Jul 16 2011

Define the infinite square array m(n,k) = (n-k)^2 for 1<=k<=n below the diagonal and m(n,k) = (k+n)(k-n) for 1<=n<=k above the diagonal. Then a(n) = Sum_{k=1..n} m(n,k) + Sum_{r=1..n} m(r,n), the "hook sum" of the terms left from m(n,n) and above m(n,n). - J. M. Bergot, Aug 16 2013

Partial sums of A049451. - Bruno Berselli, Feb 10 2014

Volume of an extruded rectangular brick with side lengths n, n and n-1. - Luciano Ancora, Jun 24 2015

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
• R. J. Mathar, On the Diophantine equation (X-Y)^m-XY=0.
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
• J.S. Seneschal, Oblong cuboid illustration.
• Eric Weisstein's World of Mathematics, Edge Count.
• Eric Weisstein's World of Mathematics, Rook Graph.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A011379, A047929, A114364 (essentially the same).

Cf. A000217, A000290, A000578, A006002, A049451, A146485.

Cf. A002378, A001105, A011379.

Cf. A028724, A052289.

Programs

Magma

[n^3-n^2: n in [0..40]]; // Vincenzo Librandi, May 02 2011

Maple

A045991:=n->n^3 - n^2: seq(A045991(n), n=0..50); # Wesley Ivan Hurt, Mar 30 2014

Mathematica

Table[n^3 - n^2, {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Dec 22 2008 *)
Table[4 Binomial[n, 2] + 6 Binomial[n, 3], {n, 0, 50}] (* Robert G. Wilson v, Mar 25 2012 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 4, 18, 48}, 20] (* Eric W. Weisstein, Jun 20 2017 *)

PARI

a(n)=n^2*(n-1) \\ Charles R Greathouse IV, Jul 17 2011

Sage

[n^2*(n-1) for n in range(0, 40)] # Zerinvary Lajos, Dec 03 2009

Formula

G.f.: 2*x^2*(x+2)/(-1+x)^4 = 6/(-1+x)^4+10/(-1+x)^2+14/(-1+x)^3+2/(-1+x). - R. J. Mathar, Nov 19 2007

a(n) = floor(n^5/(n^2+n+1)). - Gary Detlefs, Feb 10 2010

a(n) = 4*binomial(n,2) + 6*binomial(n,3). - Gary Detlefs, Mar 25 2012

a(n+1) = 2*A006002(n). - R. J. Mathar, Oct 31 2012

a(n) = (A000217(n-1)+A000217(n))*(A000217(n-1)-A000217(n-2)). - J. M. Bergot, Oct 31 2012

From Wesley Ivan Hurt, May 19 2015: (Start)

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4).

a(n) = Sum_{k=0..n-1} Sum_{i=n-k-1..n+k-1} i. (End)

Sum_{n>=2} 1/a(n) = 2 - Pi^2/6. - Daniel Suteu, Feb 06 2017

Sum_{n>=2} (-1)^n/a(n) = Pi^2/12 + 2*log(2) - 2. - Amiram Eldar, Jul 05 2020

E.g.f.: exp(x)*x^2*(2 + x). - Stefano Spezia, May 20 2021

Product_{n>=2} (1 - 1/a(n)) = A146485. - Amiram Eldar, Nov 22 2022

From J.S. Seneschal, Jun 21 2024: (Start)

a(n) = (n-1)*A000290(n).

a(n) = n*A002378(n-1).

a(n) = A011379(n) - A001105(n). (End)