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A078099
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Array T(m,n) read by antidiagonals: T(m,n) = number of ways of 3-coloring an m X n grid (m >= 1, n >= 1).
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5
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1, 2, 2, 4, 6, 4, 8, 18, 18, 8, 16, 54, 82, 54, 16, 32, 162, 374, 374, 162, 32, 64, 486, 1706, 2604, 1706, 486, 64, 128, 1458, 7782, 18150, 18150, 7782, 1458, 128, 256, 4374, 35498, 126534, 193662, 126534, 35498, 4374, 256, 512, 13122, 161926, 882180
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Half the number of mXn binary matrices with no 2X2 circuit having the pattern 0011 in any orientation [From R. H. Hardin (rhhardin(AT)att.net), Oct 06 2010]
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REFERENCES
| Michael S. Paterson (Warwick), personal communication.
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FORMULA
| Let M[1] = [1], M[m+1] = [ M[m], M[m]' / 0, M[m] ], W[m] = M[m] + M[m]', then T(m, n) = sum of entries of W[m]^(n-1) (the prime denotes transpose).
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EXAMPLE
| Array begins
1 2 4 8 16 ...
2 6 18 54 162 ...
4 18 82 374 ...
...
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MAPLE
| with(linalg); t := transpose; M[1] := matrix(1, 1, [1]); Z[1] := matrix(1, 1, 0); W[1] := evalm(M[1]+t(M[1])); v[1] := matrix(1, 1, 1);
for n from 2 to 6 do t1 := stackmatrix(M[n-1], Z[n-1]); t2 := stackmatrix(t(M[n-1]), M[n-1]); M[n] := t(stackmatrix(t(t1), t(t2))); Z[n] := matrix(2^(n-1), 2^(n-1), 0); W[n] := evalm(M[n]+t(M[n])); v[n] := matrix(1, 2^(n-1), 1); od:
T := proc(m, n) evalm( v[m] &* W[m]^(n-1) &* t(v[m]) ); end;
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CROSSREFS
| Main diagonal is A068253. Other diagonals produce A078101 and A078102. 3rd and 4th rows produce A020698 and A078100.
Sequence in context: A096466 A088965 A059474 * A118960 A107797 A038759
Adjacent sequences: A078096 A078097 A078098 * A078100 A078101 A078102
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KEYWORD
| nonn,tabl
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Dec 05 2002
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EXTENSIONS
| More terms from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Mar 23 2009
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