cp's OEIS Frontend

A364781 Triangular array read by rows: T(n, k) is the number of zero-energy states from the partition function in the Ising model for a finite n*k square lattice with periodic boundary conditions.

%I A364781 #54 Feb 01 2024 02:51:58
%S A364781 0,2,12,0,26,0,2,100,1346,20524,0,322,0,272682,0,2,1188,72824,3961300,
%T A364781 226137622,13172279424,0,4258,0,58674450,0,777714553240,0,2,15876,
%U A364781 3968690,876428620,199376325322,46463664513012,10990445640557042,2627978003957146636,0,59138,0,13184352554,0,2799323243348702,0,633566123999182005386,0
%N A364781 Triangular array read by rows: T(n, k) is the number of zero-energy states from the partition function in the Ising model for a finite n*k square lattice with periodic boundary conditions.
%C A364781 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.
%C A364781 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.
%H A364781 Manuel Kauers, <a href="/A364781/b364781.txt">Triangular array flattened. Table of n, a(n) for n = 1..120</a>
%H A364781 Roland Häggkvist, Anders Rosengren, Daniel Andrén, Petras Kundrotas, Per Håkan Lundow, and Klas Markström, <a href="http://abel.math.umu.se/~klasm/Uppsatser/plancheb.pdf">Computation of the Ising partition function for 2-dimensional square grids</a>, Phys. Rev. E 69, 046104 (April 16 2004).
%H A364781 Manuel Kauers, <a href="http://www.algebra.uni-linz.ac.at/people/mkauers/publications/kauers18k.pdf">Onsager's solution of the Ising model could have been guessed</a>, presentation slides (2018).
%H A364781 Thomas Scheuerle, <a href="/A364781/a364781.txt">Some values for T(k, k) from Klas Markström and R. Häggkvist et al.</a>, extracted from calculation results provided with their work. (See link.)
%F A364781 T(n, k) = 0 if n*k is odd.
%e A364781 Triangle begins:
%e A364781   0;
%e A364781   2,    12;
%e A364781   0,    26,       0;
%e A364781   2,   100,    1346,     20524;
%e A364781   0,   322,       0,    272682,            0;
%e A364781   2,  1188,   72824,   3961300,    226137622,    13172279424;
%e A364781   0,  4258,       0,  58674450,            0,   777714553240,                 0;
%e A364781   2, 15876, 3968690, 876428620, 199376325322, 46463664513012, 10990445640557042, 2627978003957146636;
%e A364781   ...
%o A364781 (MATLAB)
%o A364781 function a = A364781( n, k )
%o A364781     a = 0;
%o A364781     for m = 1:2^(n*k)-2
%o A364781         if isingSum( reshape(1-2*bitget(m,1:n*k),n ,k)) == 0
%o A364781             a = a + 1;
%o A364781         end
%o A364781     end
%o A364781 end
%o A364781 function e = isingSum( config )
%o A364781     e = 0; si = size(config);
%o A364781     for j = 1:si(2)
%o A364781         for k = 1:si(1)
%o A364781             S = config(k, j);
%o A364781             nb = config(1+mod(k , si(1)), j) + config(k, 1+mod(j , si(2)));
%o A364781             e = e + (-nb)*S;
%o A364781         end
%o A364781     end
%o A364781 end
%Y A364781 Cf. A001411, A002931, A010566, A241023.
%K A364781 nonn,tabl
%O A364781 1,2
%A A364781 _Thomas Scheuerle_, Aug 07 2023
%E A364781 a(27) - a(45) from _Manuel Kauers_, Sep 07 2023