A244195 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.

6, 450, 151206, 145456074, 325148366166, 1562036085226890, 17234732991509112246

Offset

1,1

Links

Table of n, a(n) for n=1..7.

Formula

Sum[P(0)>=0,P(1)>=0...P(2^n-1)>=0,

such that Sum(0<=i<=2^n-1,P(i)=n)]n!/(P(1)!P(2)!...P(2^n-1)!)

Sum[0<=b<=2^n-1]2^{P(2^n-1-b)}Prod[0<=d<=2^n-1]

(1+KroneckerDelta[0,S(d,b)](KroneckerDelta[P(d),0]-1) ,

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.

Bounded by 2^{n^2+n} < a(n) < 2^{2n^2+n}.

Example

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.

Crossrefs

Sequence in context: A265168 A187514 A258873 * A338943 A232593 A267082

Adjacent sequences: A244192 A244193 A244194 * A244196 A244197 A244198

Keyword

nonn,more

Author

Eial Teomy, Jun 22 2014

Status

approved