A341478 -id:A341478 - cp's OEIS frontend

A335586 Number of domino tilings of a 2n X 2n toroidal grid.

1, 8, 272, 90176, 311853312, 11203604497408, 4161957566985310208, 15954943354032349049274368, 630665326543010382995142219988992, 256955886436135671144699761794930161483776

Offset: 0

Drake Thomas, Jan 26 2021

nonn

For n > 1, number of perfect matchings of the graph C_2n X C_2n.

			For n = 1, there are a(1) = 8 tilings (see the Links section for a diagram).

Seiichi Manyama, Table of n, a(n) for n = 0..44
S. N. Perepechko, The number of perfect matchings on C_m X C_n graphs, (in Russian), Information Processes, 2016, V. 16, No. 4, pp. 333-361.
Drake Thomas, 8 tilings for the 2 X 2 toroidal grid.
Eric Weisstein's World of Mathematics, Perfect Matching
Eric Weisstein's World of Mathematics, Torus Grid Graph

Number of perfect matchings of the graph C_2m X C_n: A162484 (m=1), A220864 (m=2), A232804 (m=3), A253678 (m=4), A281679 (m=5), A309018 (m=6).

Cf. A004003, A212800, A340562, A341478, A341479.

default(realprecision, 120);
b(n) = round(prod(j=1, n-1, prod(k=1, n, 4*sin(j*Pi/n)^2+4*sin((2*k-1)*Pi/(2*n))^2)));
c(n) = round(prod(j=1, n, prod(k=1, n, 4*sin((2*j-1)*Pi/(2*n))^2+4*sin((2*k-1)*Pi/(2*n))^2)));
a(n) = if(n==0, 1, 4*b(n)+c(n)/2); \\ Seiichi Manyama, Feb 13 2021

a(n) = 4 * Product_{j=1..n-1} Product_{k=1..n} (4*sin(j*Pi/n)^2 + 4*sin((2*k-1)*Pi/(2*n))^2) + 1/2 * Product_{1<=j,k<=n} (4*sin((2*j-1)*Pi/(2*n))^2 + 4*sin((2*k-1)*Pi/(2*n))^2) = 4 * A341478(n)^2 + A341479(n)/2 for n > 0. - Seiichi Manyama, Feb 13 2021

a(n) ~ (1 + sqrt(2)) * exp(4*G*n^2/Pi), where G is Catalan's constant A006752. - Vaclav Kotesovec, Feb 14 2021

More terms from Seiichi Manyama, Feb 13 2021

A341479 a(n) = Product_{1<=j,k<=n} (4sin((2j-1)Pi/(2n))^2 + 4sin((2k-1)Pi/(2n))^2).

Original entry on oeis.org

1, 8, 256, 80000, 268435456, 9503683872768, 3503536769037500416, 13371518717864846127300608, 527073330112110826119518513790976, 214344906329057967318939007805581230080000

Offset: 0

Seiichi Manyama, Feb 13 2021

nonn

Cf. A335586, A340562, A341478, A341493.

Table[Product[4*Sin[(2*j - 1)*Pi/(2*n)]^2 + 4*Sin[(2*k - 1)*Pi/(2*n)]^2, {k, 1, n}, {j, 1, n}], {n, 0, 12}] // Round (* Vaclav Kotesovec, Feb 14 2021 *)

default(realprecision, 120);
a(n) = round(prod(j=1, n, prod(k=1, n, 4*sin((2*j-1)*Pi/(2*n))^2+4*sin((2*k-1)*Pi/(2*n))^2)));

a(n) ~ 2 * exp(4*G*n^2/Pi), where G is Catalan's constant A006752. - Vaclav Kotesovec, Feb 14 2021

A341535 a(n) = sqrt(Product_{1<=j,k<=n} (4sin((2j-1)Pi/(2n))^2 + 4sin((2k-1)*Pi/n)^2)).

Original entry on oeis.org

1, 2, 36, 224, 38416, 2540032, 4115479104, 3044533460992, 48656376372265216, 387018647188487143424, 62441634466575620320306176, 5221063878050546380074377019392, 8590392749565593082105293619707908096, 7476351474500749779460880888573410601336832

Offset: 0

Seiichi Manyama, Feb 13 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..62

Main diagonal of A341533.

Cf. A341478, A341479, A341782.

Table[Sqrt[Product[4*Sin[(2*j - 1)*Pi/(2*n)]^2 + 4*Sin[(2*k - 1)*Pi/n]^2, {k, 1, n}, {j, 1, n}]], {n, 0, 20}] // Round (* Vaclav Kotesovec, Feb 14 2021 *)

default(realprecision, 120);
a(n) = round(sqrt(prod(j=1, n, prod(k=1, n, 4*sin((2*j-1)*Pi/(2*n))^2+4*sin((2*k-1)*Pi/n)^2))));

a(n) ~ 2^(1/4)*(1 + sqrt(2)*(1 + (-1)^n)/2) * exp(2*G*n^2/Pi), where G is Catalan's constant A006752. - Vaclav Kotesovec, Feb 14 2021

If n is odd, a(n) = 2*A341478(n). - Seiichi Manyama, Feb 19 2021

A341782 a(n) = sqrt( Product_{j=1..n} Product_{k=1..n-1} (4sin((2j-1)Pi/(2n))^2 + 4sin(2k*Pi/n)^2) ).

Original entry on oeis.org

1, 1, 2, 112, 2312, 1270016, 292820000, 1522266730496, 3772667519238272, 193509323594243571712, 5041011532336819845120512, 2610531939025273190037188509696

Offset: 0

Seiichi Manyama, Feb 19 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..62

Main diagonal of A341738.

Cf. A341478, A341535.

Table[Sqrt[Product[Product[(4*Sin[(2*j - 1)*Pi/(2*n)]^2 + 4*Sin[2*k*Pi/n]^2), {j, 1, n}], {k, 1, n - 1}]], {n, 0, 15}] // Round (* Vaclav Kotesovec, Mar 18 2023 *)

default(realprecision, 120);
a(n) = round(sqrt(prod(j=1, n, prod(k=1, n-1, 4*sin((2*j-1)*Pi/(2*n))^2+4*sin(2*k*Pi/n)^2))));

If n is odd, a(n) = A341535(n)/2.
If n is odd, a(n) = A341478(n).
a(n) ~ exp(2*G*n^2/Pi) / (2^(3/4) * (1 + (1 + (-1)^n)/sqrt(2))), where G is Catalan's constant A006752. - Vaclav Kotesovec, Mar 18 2023

A341739 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Product_{a=1..n-1} Product_{b=1..k} (4sin(aPi/n)^2 + 4sin((2b-1)Pi/(2k))^2).

Original entry on oeis.org

1, 1, 8, 1, 36, 49, 1, 200, 625, 288, 1, 1156, 12544, 9216, 1681, 1, 6728, 279841, 583200, 130321, 9800, 1, 39204, 6385729, 44408896, 24611521, 1822500, 57121, 1, 228488, 146410000, 3546167328, 6059221281, 1003520000, 25411681, 332928

Offset: 1

Seiichi Manyama, Feb 18 2021

			Square array begins:
     1,      1,        1,          1,             1, ...
     8,     36,      200,       1156,          6728, ...
    49,    625,    12544,     279841,       6385729, ...
   288,   9216,   583200,   44408896,    3546167328, ...
  1681, 130321, 24611521, 6059221281, 1612940640256, ...

Main diagonal gives A341478(n)^2.

Cf. A341533, A341738, A341741.

default(realprecision, 120);
T(n, k) = round(prod(a=1, n-1, prod(b=1, k, 4*sin(a*Pi/n)^2+4*sin((2*b-1)*Pi/(2*k))^2)));

cp's OEIS Frontend

A335586 Number of domino tilings of a 2n X 2n toroidal grid.

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A341479 a(n) = Product_{1<=j,k<=n} (4*sin((2*j-1)*Pi/(2*n))^2 + 4*sin((2*k-1)*Pi/(2*n))^2).

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A341535 a(n) = sqrt(Product_{1<=j,k<=n} (4*sin((2*j-1)*Pi/(2*n))^2 + 4*sin((2*k-1)*Pi/n)^2)).

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A341782 a(n) = sqrt( Product_{j=1..n} Product_{k=1..n-1} (4*sin((2*j-1)*Pi/(2*n))^2 + 4*sin(2*k*Pi/n)^2) ).

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Author

Keywords

Links

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Programs

Mathematica

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Formula

A341739 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Product_{a=1..n-1} Product_{b=1..k} (4*sin(a*Pi/n)^2 + 4*sin((2*b-1)*Pi/(2*k))^2).

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Examples

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Programs

PARI

A341479 a(n) = Product_{1<=j,k<=n} (4sin((2j-1)Pi/(2n))^2 + 4sin((2k-1)Pi/(2n))^2).

A341535 a(n) = sqrt(Product_{1<=j,k<=n} (4sin((2j-1)Pi/(2n))^2 + 4sin((2k-1)*Pi/n)^2)).

A341782 a(n) = sqrt( Product_{j=1..n} Product_{k=1..n-1} (4sin((2j-1)Pi/(2n))^2 + 4sin(2k*Pi/n)^2) ).

A341739 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Product_{a=1..n-1} Product_{b=1..k} (4sin(aPi/n)^2 + 4sin((2b-1)Pi/(2k))^2).