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).

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

Offset: 1

Author

N. J. A. Sloane, Dec 05 2002

Keywords

• nonn
• tabl

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

Examples

			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
		

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

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).

Programs

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 *)

Formula

Let M[1] = [1], M[m+1] = the block matrix [ [ 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

Extensions

More terms from Alois P. Heinz, Mar 23 2009