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a(n) = n^3 - n^2.
64

%I #176 Jun 27 2024 03:53:25

%S 0,0,4,18,48,100,180,294,448,648,900,1210,1584,2028,2548,3150,3840,

%T 4624,5508,6498,7600,8820,10164,11638,13248,15000,16900,18954,21168,

%U 23548,26100,28830,31744,34848,38148,41650,45360,49284,53428,57798,62400,67240,72324

%N a(n) = n^3 - n^2.

%C 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

%C Number of edges of the Cartesian product of two complete graphs K_n X K_n. - _Roberto E. Martinez II_, Jan 07 2002

%C That is, number of edges in the n X n rook graph. - _Eric W. Weisstein_, Jun 20 2017

%C n such that x^3 + x^2 + n factors over the integers. - _James R. Buddenhagen_, Apr 19 2005

%C 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

%C 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

%C 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

%C a(n) equals the coefficient of log(2) in 2F1(n-1,n-1,n+1,-1). - _John M. Campbell_, Jul 16 2011

%C 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

%C Partial sums of A049451. - _Bruno Berselli_, Feb 10 2014

%C Volume of an extruded rectangular brick with side lengths n, n and n-1. - _Luciano Ancora_, Jun 24 2015

%H Vincenzo Librandi, <a href="/A045991/b045991.txt">Table of n, a(n) for n = 0..1000</a>

%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>.

%H R. J. Mathar, <a href="/A045991/a045991.pdf">On the Diophantine equation (X-Y)^m-XY=0</a>.

%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EdgeCount.html">Edge Count</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F 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

%F a(n) = floor(n^5/(n^2+n+1)). - _Gary Detlefs_, Feb 10 2010

%F a(n) = 4*binomial(n,2) + 6*binomial(n,3). - _Gary Detlefs_, Mar 25 2012

%F a(n+1) = 2*A006002(n). - _R. J. Mathar_, Oct 31 2012

%F a(n) = (A000217(n-1)+A000217(n))*(A000217(n-1)-A000217(n-2)). - _J. M. Bergot_, Oct 31 2012

%F From _Wesley Ivan Hurt_, May 19 2015: (Start)

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

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

%F Sum_{n>=2} 1/a(n) = 2 - Pi^2/6. - _Daniel Suteu_, Feb 06 2017

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

%F E.g.f.: exp(x)*x^2*(2 + x). - _Stefano Spezia_, May 20 2021

%F Product_{n>=2} (1 - 1/a(n)) = A146485. - _Amiram Eldar_, Nov 22 2022

%F From _J.S. Seneschal_, Jun 21 2024: (Start)

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

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

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

%p A045991:=n->n^3 - n^2: seq(A045991(n), n=0..50); # _Wesley Ivan Hurt_, Mar 30 2014

%t Table[n^3 - n^2, {n, 0, 50}] (* _Vladimir Joseph Stephan Orlovsky_, Dec 22 2008 *)

%t Table[4 Binomial[n, 2] + 6 Binomial[n, 3], {n, 0, 50}] (* _Robert G. Wilson v_, Mar 25 2012 *)

%t LinearRecurrence[{4, -6, 4, -1}, {0, 4, 18, 48}, 20] (* _Eric W. Weisstein_, Jun 20 2017 *)

%o (Sage) [n^2*(n-1) for n in range(0, 40)] # _Zerinvary Lajos_, Dec 03 2009

%o (Magma) [n^3-n^2: n in [0..40]]; // _Vincenzo Librandi_, May 02 2011

%o (PARI) a(n)=n^2*(n-1) \\ _Charles R Greathouse IV_, Jul 17 2011

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

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

%Y Cf. A002378, A001105, A011379.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_