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A078099 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). 12
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, 2068146, 2068146, 882180, 161926, 13122, 512 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
We assume the top left point gets color 1 (or, in other words, take the total number of colorings and divide by 3). The rule for coloring is that horizontally or vertically adjacent points must have different colors. - N. J. A. Sloane, Feb 12 2013
Equals half the number of m X n binary matrices with no 2 X 2 circuit having the pattern 0011 in any orientation. - R. H. Hardin, Oct 06 2010
Also the number of Miura-ori foldings [Ginepro-Hull]. - N. J. A. Sloane, Aug 05 2015
REFERENCES
Thomas C. Hull, Coloring Connections with Counting Mountain-Valley Assignments in (book) Origami^6: I. Mathematics, 2015, ed. Koryo Miura, Toshikazu Kawasaki, Tomohiro Tachi, Ryuhei Uehara, Robert J. Lang, Patsy Wang-Iverson, American Mathematical Soc., Dec 18, 2015, 368 pages
Michael S. Paterson (Warwick), personal communication.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1128 (terms 1..120 from R. J. Mathar)
J. Ginepro, T. C. Hull, Counting Miura-ori Foldings, Journal of Integer Sequences, Vol. 17, 2014, #14.10.8
R. J. Mathar, Counting 2-way monotonic terrace forms..., vixra 1511.0225 (2015), Table T_{n x m}
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Vertex Coloring
Wikipedia, Graph Coloring
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).
T(3,n) = A052913(n). T(4,n) = 2*A078100(n).
T(n,m) = T(m,n). T(1,n)= A000079(n-1). T(2,n)=A025192(n). T(5,n) = 2*A207994(n). T(6,n) = 2*A207995(n). - R. J. Mathar, Nov 23 2015
EXAMPLE
Array begins:
1 2 4 8 16 32 64 128 256 512 ...
2 6 18 54 162 486 1458 4374 13122 ...
4 18 82 374 1706 7782 35498 161926 ...
8 54 374 2604 18150 126534 882180 ...
16 162 1706 18150 193662 ...
32 486 7782 126534 ...
For the 1 X n case: the first point gets color 1, thereafter there are 2 choices for each color, so T(1,n) = 2^(n-1).
For the 2 X 2 case, the colorings are
12 12 12 13 13 13
21 23 31 31 32 21
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;
MATHEMATICA
mmax = 10; M[1] = {{1}}; M[m_] := M[m] = {{M[m-1], Transpose[M[m-1]]}, {Array[0&, {2^(m-2), 2^(m-2)}], M[m-1]}} // ArrayFlatten; W[m_] := M[m] + Transpose[M[m]]; T[m_, 1] := 2^(m-1); T[1, n_] := 2^(n-1); T[m_, n_] := MatrixPower[W[m], n-1] // Flatten // Total; Table[T[m-n+1, n], {m, 1, mmax}, {n, 1, m}] // Flatten (* Jean-François Alcover, Feb 13 2016 *)
CROSSREFS
Cf. A207997, A020698, A078100. Main diagonal is A068253. Other diagonals produce A078101 and A078102.
Cf. A222444 (4 colorings), A222144 (5 colorings), A222281 (6 colorings), A222340 (7 colorings), A222462 (8 colorings).
Sequence in context: A059474 A252828 A208314 * A264872 A303306 A347797
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Dec 05 2002
EXTENSIONS
More terms from Alois P. Heinz, Mar 23 2009
STATUS
approved

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Last modified July 21 03:47 EDT 2024. Contains 374463 sequences. (Running on oeis4.)