Domino Tiling

Author

Eric W. Weisstein
August 31, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/DominoTiling.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/DominoTiling.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Tiling

Distinct Dimer Coverings on an N==n×n Square, n Even

(2^((n^2)/2)) (∏_(k=1)^(n/2))(∏_(j=1)^(n/2))((cos^2)(((j π)/(n+1)))+(cos^2)(((k π)/(n+1))))

(2^((n^2)/2))Product[((Cos[((j π)/(n+1))])^2)+((Cos[((k π)/(n+1))])^2),{k,n/2},{j,n/2}]

Product[((Cos[((j π)/(2nn+1))])^2)+((Cos[((k π)/(2nn+1))])^2),{k,nn}]

l0=Table[(2^((n^2)/2))Product[((Cos[((j π)/(n+1))])^2)+((Cos[((k π)/(n+1))])^2),{k,n/2},{j,n/2}],{n,2,26,2}];

l0=Table[(2^(2(n^2)))Product[((Cos[((j π)/(2n+1))])^2)+((Cos[((k π)/(2n+1))])^2),{k,n},{j,n}],{n,15}];

http://www.research.att.com/~njas/sequences/A004003

l=l0//TrigToExp//Simplify//RootReduce

l=l0//TrigToExp//Simplify//FullSimplify

l=l0//Round

Figure

4×4

tiling={{{1,2},{3,4},{5,6},{7,8},{9,10},{11,12},{13,14},{15,16}},{{1,2},{3,4},{5,6},{7,11},{9,10},{8,12},{13,14},{15,16}},{{1,2},{3,4},{5,9},{6,7},{10,11},{8,12},{13,14},{15,16}},{{1,2},{3,4},{5,9},{6,10},{7,8},{11,12},{13,14},{15,16}},{{1,2},{3,4},{5,9},{6,10},{7,11},{8,12},{13,14},{15,16}},{{1,2},{3,4},{5,6},{7,8},{9,10},{13,14},{11,15},{12,16}},{{1,2},{3,4},{5,9},{6,10},{7,8},{11,15},{13,14},{12,16}},{{1,2},{3,4},{5,6},{7,8},{9,13},{10,14},{11,12},{15,16}},{{1,2},{3,4},{5,6},{7,11},{8,12},{9,13},{10,14},{15,16}},{{1,2},{3,4},{5,6},{7,8},{9,13},{10,11},{14,15},{12,16}},{{1,2},{3,4},{5,6},{7,8},{9,13},{10,14},{11,15},{12,16}},{{1,2},{5,6},{3,7},{4,8},{9,10},{11,12},{13,14},{15,16}},{{1,2},{3,7},{4,8},{5,9},{6,10},{11,12},{13,14},{15,16}},{{1,2},{5,6},{3,7},{4,8},{9,10},{13,14},{11,15},{12,16}},{{1,2},{3,7},{4,8},{6,10},{5,9},{11,15},{12,16},{13,14}},{{1,2},{3,7},{4,8},{5,6},{9,13},{10,14},{11,12},{15,16}},{{1,2},{3,7},{4,8},{5,6},{9,13},{10,11},{14,15},{12,16}},{{1,2},{5,6},{3,7},{4,8},{9,13},{10,14},{11,15},{12,16}},{{1,5},{2,6},{3,4},{7,8},{9,10},{11,12},{13,14},{15,16}},{{1,5},{2,6},{3,4},{7,11},{8,12},{9,10},{13,14},{15,16}},{{1,5},{3,4},{2,6},{9,10},{7,8},{11,15},{13,14},{12,16}},{{1,5},{2,6},{3,4},{7,8},{9,13},{10,14},{11,12},{15,16}},{{1,5},{2,6},{3,4},{7,11},{8,12},{9,13},{10,14},{15,16}},{{1,5},{2,6},{3,4},{7,8},{9,13},{10,11},{14,15},{12,16}},{{1,5},{2,6},{3,4},{7,8},{9,13},{10,14},{11,15},{12,16}},{{1,5},{2,3},{6,7},{4,8},{9,10},{11,12},{13,14},{15,16}},{{1,5},{2,6},{3,7},{4,8},{9,10},{11,12},{13,14},{15,16}},{{1,5},{2,3},{6,7},{4,8},{9,10},{11,15},{13,14},{12,16}},{{1,5},{2,6},{3,7},{4,8},{9,10},{13,14},{11,15},{12,16}},{{1,5},{2,3},{6,7},{4,8},{9,13},{10,14},{11,12},{15,16}},{{1,5},{2,6},{3,7},{4,8},{9,13},{10,14},{11,12},{15,16}},{{1,5},{2,6},{3,7},{4,8},{9,13},{10,14},{11,15},{12,16}},{{1,5},{2,6},{3,7},{4,8},{9,13},{10,11},{14,15},{12,16}},{{1,5},{2,3},{4,8},{6,10},{7,11},{9,13},{14,15},{12,16}},{{1,5},{2,3},{6,7},{4,8},{9,13},{10,14},{11,15},{12,16}},{{1,5},{2,3},{6,7},{4,8},{9,13},{10,11},{14,15},{12,16}}};

t=Flatten/@MapIndexed[Flatten[Thread[(#1/.n_Integer:>{1+Mod[n-1,4],1+Floor[(n-1)/4]})->#2[[2]]]]&,tiling,{2}];

GraphicsGrid[Partition[ArrayPlot[SparseArray[#],ColorFunction->Hue]&/@t,8,8,{1,1},{}],ImageSize->400]

Mod Relation

(Subscript[A, n])=(2^n)((Subscript[B, n])^2)

Mod[MapIndexed[Sqrt[#1/2^(#2[[1]])]&,l],32]

Mod[Table[If[EvenQ[n],n+1,(((-1))^((n-1)/2))n],{n,20}],32]

Table[2^n ((If[OddQ[n],(-1)^((n-1)/2)n,n+1])^2),{n,20}]

Limit

N[Catalan/Pi,10]

MapIndexed[((Log[#])/(((2#2[[1]]))^2))&,l]//N

SequenceLimit[%]

Integral

(1/(16(π^2)))NIntegrate[Log[4+2Cos[θ]+2Cos[ϕ]],{θ,-π,π},{ϕ,-π,π}]

(1/(16(π^2)))Integrate[Integrate[Log[4+2Cos[θ]+2Cos[ϕ]],{θ,-π,π},Assumptions->-π<ϕ<π],{ϕ,-π,π}]//Timing

(1/(16(π^2)))Integrate[Log[4+2Cos[θ]+2Cos[ϕ]],{θ,-π,π},{ϕ,-π,π}]//Timing

(1/(16 (π^2)))NIntegrate[-I ((π^2)+4 π ArcSin[((Sqrt[3+Cos[θ]])/(Sqrt[2]))]+4 I ArcSin[((Sqrt[3+Cos[θ]])/(Sqrt[2]))] Log[3+Cos[θ]-(Sqrt[3+4 Cos[θ]+((Cos[θ])^2)])]+4 I ArcSin[((Sqrt[3+Cos[θ]])/(Sqrt[2]))] Log[-((1+Cos[θ]+(Sqrt[3+4 Cos[θ]+((Cos[θ])^2)]))/(-1-Cos[θ]+(Sqrt[3+4 Cos[θ]+((Cos[θ])^2)])))]-4 I ArcSin[((Sqrt[3+Cos[θ]])/(Sqrt[2]))] Log[3+Cos[θ]+(Sqrt[3+4 Cos[θ]+((Cos[θ])^2)])]+2 PolyLog[2,-2-Cos[θ]-(Sqrt[3+4 Cos[θ]+((Cos[θ])^2)])]-2 PolyLog[2,2+Cos[θ]-(Sqrt[3+4 Cos[θ]+((Cos[θ])^2)])]+2 PolyLog[2,-2-Cos[θ]+(Sqrt[3+4 Cos[θ]+((Cos[θ])^2)])]-2 PolyLog[2,2+Cos[θ]+(Sqrt[3+4 Cos[θ]+((Cos[θ])^2)])]),{θ,-π,π}]

FullSimplify[-I ((π^2)+4 π ArcSin[((Sqrt[3+Cos[θ]])/(Sqrt[2]))]+4 I ArcSin[((Sqrt[3+Cos[θ]])/(Sqrt[2]))] Log[3+Cos[θ]-(Sqrt[3+4 Cos[θ]+((Cos[θ])^2)])]+4 I ArcSin[((Sqrt[3+Cos[θ]])/(Sqrt[2]))] Log[-((1+Cos[θ]+(Sqrt[3+4 Cos[θ]+((Cos[θ])^2)]))/(-1-Cos[θ]+(Sqrt[3+4 Cos[θ]+((Cos[θ])^2)])))]-4 I ArcSin[((Sqrt[3+Cos[θ]])/(Sqrt[2]))] Log[3+Cos[θ]+(Sqrt[3+4 Cos[θ]+((Cos[θ])^2)])]+2 PolyLog[2,-2-Cos[θ]-(Sqrt[3+4 Cos[θ]+((Cos[θ])^2)])]-2 PolyLog[2,2+Cos[θ]-(Sqrt[3+4 Cos[θ]+((Cos[θ])^2)])]+2 PolyLog[2,-2-Cos[θ]+(Sqrt[3+4 Cos[θ]+((Cos[θ])^2)])]-2 PolyLog[2,2+Cos[θ]+(Sqrt[3+4 Cos[θ]+((Cos[θ])^2)])]),0<θ<π]

(2/(16 (π^2)))NIntegrate[-I ((π^2)+2 PolyLog[2,-2-Cos[θ]-(Sqrt[(1+Cos[θ]) (3+Cos[θ])])]-2 PolyLog[2,2+Cos[θ]-(Sqrt[(1+Cos[θ]) (3+Cos[θ])])]+2 PolyLog[2,-2-Cos[θ]+(Sqrt[(1+Cos[θ]) (3+Cos[θ])])]-2 PolyLog[2,2+Cos[θ]+(Sqrt[(1+Cos[θ]) (3+Cos[θ])])]),{θ,0,π}]

expr=-I ((π^2)+2 PolyLog[2,-2-Cos[θ]-(Sqrt[(1+Cos[θ]) (3+Cos[θ])])]-2 PolyLog[2,2+Cos[θ]-(Sqrt[(1+Cos[θ]) (3+Cos[θ])])]+2 PolyLog[2,-2-Cos[θ]+(Sqrt[(1+Cos[θ]) (3+Cos[θ])])]-2 PolyLog[2,2+Cos[θ]+(Sqrt[(1+Cos[θ]) (3+Cos[θ])])]);

Integrate[expr,{θ,0,π}]

Solve[(2/(16 (π^2)))i==Catalan/Pi,i]

NIntegrate[expr,{θ,0,π}]

8.π Catalan