A103997 - OEIS

A103997

Square array T(M,N) read by antidiagonals: number of dimer tilings of a 2*M X 2*N Moebius strip.

1, 1, 1, 1, 3, 1, 1, 11, 7, 1, 1, 41, 71, 18, 1, 1, 153, 769, 539, 47, 1, 1, 571, 8449, 17753, 4271, 123, 1, 1, 2131, 93127, 603126, 434657, 34276, 322, 1, 1, 7953, 1027207, 20721019, 46069729, 10894561, 276119, 843, 1, 1, 29681, 11332097, 714790675, 4974089647, 3625549353, 275770321, 2226851, 2207, 1

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OFFSET

0,5

LINKS

Table of n, a(n) for n=0..54.

Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter and Tianyuan Xu, Sandpiles and Dominos, El. J. Comb., 22 (2015), P1.66. See Theorem 18.

W. T. Lu and F. Y. Wu, Dimer statistics on the Moebius strip and the Klein bottle, arXiv:cond-mat/9906154 [cond-mat.stat-mech], 1999.

Index entries for sequences related to dominoes

FORMULA

T(M, N) = Product_{m=1..M} (Product_{n=1..N} 4*sin(Pi*(4*n-1)/(4*N))^2 + 4*cos(Pi*m/(2*M + 1))^2).

For k > 0, T(n,k) = 2^n * sqrt(Resultant(U_{2*n}(x/2), T_{2*k}(i*x/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1). - Seiichi Manyama, Apr 15 2020

EXAMPLE

Array begins:

1, 1, 1, 1, 1, 1, 1,

1, 3, 7, 18, 47, 123, 322,

1, 11, 71, 539, 4271, 34276, 276119,

1, 41, 769, 17753, 434657, 10894561, 275770321,

1, 153, 8449, 603126, 46069729, 3625549353, 289625349454,

1, 571, 93127, 20721019, 4974089647, 1234496016491, 312007855309063,

...

MATHEMATICA

T[M_, N_] := Product[4Sin[(4n-1)Pi/(4N)]^2 + 4Cos[m Pi/(2M+1)]^2, {n, 1, N}, {m, 1, M}];

Table[T[M - N, N] // Round, {M, 0, 9}, {N, 0, M}] // Flatten (* Jean-François Alcover, Dec 03 2018 *)