OFFSET
1,2
COMMENTS
Imagine an n X k square tiling on a 2D surface with torus topology. T(n, k) is the number of ways two colors can be assigned to all tiles such that the overall length of the boundary between the colored regions is n*k.
The number of solutions with the additional constrain that exactly k tiles must have the lesser represented color is given for tilings with size 2 X 2*k by A241023(k). In the case 2 X 2*k is k also the minimum count of tiles with the same color in all solutions.
LINKS
Manuel Kauers, Triangular array flattened. Table of n, a(n) for n = 1..120
Roland Häggkvist, Anders Rosengren, Daniel Andrén, Petras Kundrotas, Per Håkan Lundow, and Klas Markström, Computation of the Ising partition function for 2-dimensional square grids, Phys. Rev. E 69, 046104 (April 16 2004).
Manuel Kauers, Onsager's solution of the Ising model could have been guessed, presentation slides (2018).
Thomas Scheuerle, Some values for T(k, k) from Klas Markström and R. Häggkvist et al., extracted from calculation results provided with their work. (See link.)
FORMULA
T(n, k) = 0 if n*k is odd.
EXAMPLE
Triangle begins:
0;
2, 12;
0, 26, 0;
2, 100, 1346, 20524;
0, 322, 0, 272682, 0;
2, 1188, 72824, 3961300, 226137622, 13172279424;
0, 4258, 0, 58674450, 0, 777714553240, 0;
2, 15876, 3968690, 876428620, 199376325322, 46463664513012, 10990445640557042, 2627978003957146636;
...
PROG
(MATLAB)
function a = A364781( n, k )
a = 0;
for m = 1:2^(n*k)-2
if isingSum( reshape(1-2*bitget(m, 1:n*k), n , k)) == 0
a = a + 1;
end
end
end
function e = isingSum( config )
e = 0; si = size(config);
for j = 1:si(2)
for k = 1:si(1)
S = config(k, j);
nb = config(1+mod(k , si(1)), j) + config(k, 1+mod(j , si(2)));
e = e + (-nb)*S;
end
end
end
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Thomas Scheuerle, Aug 07 2023
EXTENSIONS
a(27) - a(45) from Manuel Kauers, Sep 07 2023
STATUS
approved