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A103999 Square array T(M,N) read by antidiagonals: number of dimer tilings of a 2M x 2N Klein bottle. 6
1, 1, 1, 1, 6, 1, 1, 16, 34, 1, 1, 54, 196, 198, 1, 1, 196, 1666, 2704, 1154, 1, 1, 726, 16384, 64152, 37636, 6726, 1, 1, 2704, 171394, 1844164, 2549186, 524176, 39202, 1, 1, 10086, 1844164, 57523158, 220581904, 101757654, 7300804, 228486, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Table of n, a(n) for n=0..44.

Cliff, Danny and Zoe Stoll, About Klein bottles

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

FORMULA

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

EXAMPLE

Array begins:

  1,   1,     1,        1,           1,             1,                1, ...

  1,   6,    34,      198,        1154,          6726,            39202, ...

  1,  16,   196,     2704,       37636,        524176,          7300804, ...

  1,  54,  1666,    64152,     2549186,     101757654,       4064620168, ...

  1, 196, 16384,  1844164,   220581904,   26743369156,    3252222705664, ...

  1, 726,171394, 57523158, 21050622914, 7902001927776, 2988827208115522, ...

MATHEMATICA

T[m_, n_] := Product[4 Sin[(4k-1) Pi/(4n)]^2 + 4 Cos[j Pi/(2m+1)]^2, {j, 1, m}, {k, 1, n}] // Round;

Table[T[m-n, n], {m, 0, 9}, {n, 0, m}] // Flatten (* Jean-Fran├žois Alcover, Aug 20 2018 *)

PROG

(PARI) default(realprecision, 120);

{T(n, k) = round(prod(a=1, n, prod(b=1, k, 4*sin((4*a-1)*Pi/(4*n))^2+4*sin((2*b-1)*Pi/(2*k))^2)))} \\ Seiichi Manyama, Jan 11 2021

CROSSREFS

Rows include A003499, A067902+2. Columns include A003500+2.

Main diagonal gives A340557.

Cf. A099390, A103997.

Sequence in context: A176560 A152602 A119726 * A154985 A157275 A157268

Adjacent sequences:  A103996 A103997 A103998 * A104000 A104001 A104002

KEYWORD

nonn,tabl

AUTHOR

Ralf Stephan, Feb 26 2005

STATUS

approved

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Last modified March 4 20:38 EST 2021. Contains 341803 sequences. (Running on oeis4.)