%I A045991
%S A045991 0,0,4,18,48,100,180,294,448,648,900,1210,1584,2028,2548,3150,3840,
%T A045991 4624,5508,6498,7600,8820,10164,11638,13248,15000,16900,18954,21168,
%U A045991 23548,26100,28830,31744,34848,38148,41650,45360,49284,53428,57798
%N A045991 n^3-n^2.
%C A045991 Number of edges in the line graph of the complete bipartite graph of 
               order 2n, L(K_n,n) - Roberto E. Martinez II (remartin(AT)fas.harvard.edu), 
               Jan 07 2002
%C A045991 Number of edges of the product of two complete graphs, each of order 
               n, K_n x K_n - Roberto E. Martinez II (remartin(AT)fas.harvard.edu), 
               Jan 07 2002
%C A045991 n such that x^3 + x^2 + n factors over the integers. - James Buddenhagen 
               (jbuddenh(AT)gmail.com), Apr 19 2005
%C A045991 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 (joshua.zucker(AT)stanfordalumni.org), Jan 11 
               2006
%C A045991 Sequence allows us to find X values of 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,...than the equation 
               (X-Y)^m-XY=0,... has no solution. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), 
               May 10 2006
%C A045991 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 R. Janjic (agnus(AT)blic.net), 
               Mar 13 2007
%C A045991 Number of units of a(n) belongs to a periodic sequence: 0, 0, 4, 8, 8. 
               [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 04 2009]
%H A045991 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A045991 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas 
               for Some Functions on Finite Sets</a>
%H A045991 R. J. Mathar, <a href="http://www.strw.leidenuniv.nl/~mathar/public/mathar20070502.pdf">
               On the Diophantine equation (X-Y)^m-XY=0</a> (PDF).
%F A045991 a(n)=sum(sum(n, j=2..n),k=1..n): n>=0. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               May 11 2007
%F A045991 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 (mathar(AT)strw.leidenuniv.nl), Nov 19 2007
%p A045991 a:=n->sum(numbperm (n,2), j=1..n): seq(a(n), n=0..39); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Apr 25 2007
%p A045991 a:=n->sum(sum(n, j=2..n),k=1..n): seq(a(n), n=0..39); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), May 11 2007
%p A045991 a:=n->sum(sum(sum(1, j=1..n), k=0..n),m=0..n): seq(a(n), n=-1..38); - 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 18 2007
%t A045991 a[n_]:=n^3-n^2;lst={};Do[AppendTo[lst,a[n]],{n,0,5!}];lst [From Vladimir 
               Orlovsky (4vladimir(AT)gmail.com), Dec 22 2008]
%o A045991 (Other) sage: [n^2*(n-1) for n in xrange(0, 40)] # [From Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Dec 03 2009]
%Y A045991 Cf. A047929.
%Y A045991 Cf. A011379.
%Y A045991 Sequence in context: A066153 A023650 A163188 this_sequence A114364 A027271 
               A073991
%Y A045991 Adjacent sequences: A045988 A045989 A045990 this_sequence A045992 A045993 
               A045994
%K A045991 nonn,new
%O A045991 0,3
%A A045991 N. J. A. Sloane (njas(AT)research.att.com).