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%I #12 Apr 18 2022 10:14:23
%S 6,450,151206,145456074,325148366166,1562036085226890,
%T 17234732991509112246
%N Number of ways to arrange 3n^3 spins in an n^3 cubic lattice such that each site contains 3 spins, two of which are the same, and each spin is the same as the spins in the neighboring sites in the proper direction.
%F Sum[P(0)>=0,P(1)>=0...P(2^n-1)>=0,
%F such that Sum(0<=i<=2^n-1,P(i)=n)]n!/(P(1)!P(2)!...P(2^n-1)!)
%F Sum[0<=b<=2^n-1]2^{P(2^n-1-b)}Prod[0<=d<=2^n-1]
%F (1+KroneckerDelta[0,S(d,b)](KroneckerDelta[P(d),0]-1) ,
%F where S(d,b)=0 iff d and b as written in binary digits have two identical pairs of different value. For example b=5 (101) and d=4 (100) have this property because the third digits and the second digits are the same in both of them, but they are of different value.
%F Bounded by 2^{n^2+n} < a(n) < 2^{2n^2+n}.
%e For n=2, there are 8 sites in the 2x2x2 cube. Inside each site there are three spins (0 or 1) each pointing in a different direction x,y or z. The x-spin in site 1-1-1 has the same value as the x-spin in site 2-1-1 (since they are neighbors in the x direction). The y-spin in site 1-1-1 has the same value as the y-spin in site 1-2-1, etc. Amongst the three spins in site 1-1-1, either two of them are 0 and the other is 1, or two of them are 1 and the other is 0.
%K nonn,more
%O 1,1
%A _Eial Teomy_, Jun 22 2014
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