A340562 -id:A340562 - 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

A212800 Number of spanning trees of the (n,n)-torus grid graph.

Original entry on oeis.org

1, 32, 11664, 42467328, 1562500000000, 587312954081280000, 2266101334892340404752384, 89927963805390785392395474173952, 36735015407753190053984060991247792275456, 154528563849617762057150663767149772800000000000000, 6695315138840257072470706538467584763944601124280722177130496

Offset: 1

N. J. A. Sloane, May 27 2012

nonn

Main diagonal of array in A212796.

Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212. See p. 210, Para. 4.
Eric Weisstein's World of Mathematics, Spanning Tree
Eric Weisstein's World of Mathematics, Torus Grid Graph

Cf. A212796, A340562.

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

a(n) ~ Gamma(1/4)^4 * exp(4*G*n^2/Pi) / (16 * Pi^3), where G is Catalan's constant A006752. - Vaclav Kotesovec, Feb 14 2021

More terms from Eric W. Weisstein, May 10 2017

A341478 a(n) = sqrt( Product_{1<=j<=n-1} Product_{1<=k<=n} (4sin(jPi/n)^2 + 4sin((2k-1)Pi/(2n))^2) ).

Original entry on oeis.org

1, 1, 6, 112, 6664, 1270016, 776239200, 1522266730496, 9580300901941376, 193509323594243571712, 12545297912843041612924416, 2610531939025273190037188509696, 1743627211475190637398673259679582208

Offset: 0

Seiichi Manyama, Feb 13 2021

nonn

Cf. A335586, A340562, A341479, A341493.

Table[Sqrt[Product[4*Sin[j*Pi/n]^2 + 4*Sin[(2*k - 1)*Pi/(2*n)]^2, {k, 1, n}, {j, 1, n-1}]], {n, 0, 15}] // Round (* Vaclav Kotesovec, Feb 14 2021 *)

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

a(n) ~ exp(2*G*n^2/Pi) / 2^(3/4), where G is Catalan's constant A006752. - Vaclav Kotesovec, Feb 14 2021

A229728 Decimal expansion of the square of the constant A130834.

Original entry on oeis.org

3, 2, 0, 9, 9, 1, 2, 3, 0, 0, 7, 2, 8, 1, 5, 7, 6, 7, 8, 6, 2, 9, 7, 4, 9, 4, 8, 1, 7, 7, 9, 9, 0, 5, 1, 5, 8, 7, 4, 8, 5, 9, 2, 1, 2, 4, 2, 5, 1, 8, 3, 4, 4, 9, 4, 8, 7, 4, 5, 8, 6, 0, 0, 5, 8, 4, 6, 1, 0, 2, 4, 6, 4, 1, 6, 2, 4, 2, 4, 0, 2, 0, 4, 0, 6, 6, 7, 6, 7, 1, 2, 1, 5, 1, 4, 1, 0, 8, 8, 7, 0, 9, 4, 2, 8, 4, 6, 6, 9, 1, 5, 8, 3, 8, 7, 5, 2, 2, 6, 9

Offset: 1

N. J. A. Sloane, Oct 01 2013

			3.209912300728157678629749481779905158748592124251834494874586...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 232.

Cf. A000796, A006752, A097469, A130834.

Cf. A007341, A212800, A334088, A335586, A340139, A340176, A340557, A340562, A341479.

RealDigits[Exp[4*Catalan/Pi], 10, 120][[1]] (* Amiram Eldar, Jun 12 2023 *)

exp(4*Catalan/Pi) \\ Charles R Greathouse IV, Jul 15 2014

From Amiram Eldar, Jun 12 2023: (Start)
Equals exp(4*G/Pi) = exp(4*A006752/A000796).
Equals A097469^4. (End)

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

A340560 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-1} (4sin(aPi/n)^2 + 4sin(bPi/k)^2).

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 49, 49, 1, 1, 288, 1296, 288, 1, 1, 1681, 30625, 30625, 1681, 1, 1, 9800, 707281, 2654208, 707281, 9800, 1, 1, 57121, 16257024, 219069601, 219069601, 16257024, 57121, 1, 1, 332928, 373301041, 17860500000, 62500000000, 17860500000, 373301041, 332928, 1

Offset: 1

Seiichi Manyama, Jan 11 2021

			Square array begins:
  1,    1,      1,         1,           1, ...
  1,    8,     49,       288,        1681, ...
  1,   49,   1296,     30625,      707281, ...
  1,  288,  30625,   2654208,   219069601, ...
  1, 1681, 707281, 219069601, 62500000000, ...

Rows and columns 1..2 give A000012, A001108.
Main diagonal gives A340562.
Cf. A212796, A340475, A340561.

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

T(n,k) = T(k,n).

T(n,k) = A212796(n,k)/(n*k).

A340563 a(n) = sqrt( Product_{1<=j, k<=n-1} (4sin(jPi/n)^2 + 4cos(kPi/n)^2) ).

Original entry on oeis.org

1, 1, 2, 16, 384, 30976, 7741440, 6369316864, 16435095011328, 138915523039657984, 3696387867279360000000, 321533678904455375050768384, 88192375153215003517412966400000, 78996127242669742603293261855977373696, 223311937686075869460797609709638544686841856

Offset: 0

Seiichi Manyama, Jan 11 2021

nonn

Main diagonal of A340561.

Cf. A127606, A340562.

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

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

a(n) ~ c * (sqrt(2) - 1)^n * exp(2*G*n^2/Pi), where c = sqrt(Pi) / Gamma(3/4)^2 if n is even and c = 2^(1/4) if n is odd, G is Catalan's constant A006752. - Vaclav Kotesovec, Mar 18 2023

cp's OEIS Frontend

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

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A212800 Number of spanning trees of the (n,n)-torus grid graph.

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A341478 a(n) = sqrt( Product_{1<=j<=n-1} Product_{1<=k<=n} (4*sin(j*Pi/n)^2 + 4*sin((2*k-1)*Pi/(2*n))^2) ).

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A229728 Decimal expansion of the square of the constant A130834.

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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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A340560 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-1} (4*sin(a*Pi/n)^2 + 4*sin(b*Pi/k)^2).

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A340563 a(n) = sqrt( Product_{1<=j, k<=n-1} (4*sin(j*Pi/n)^2 + 4*cos(k*Pi/n)^2) ).

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Programs

Mathematica

PARI

Formula

A341478 a(n) = sqrt( Product_{1<=j<=n-1} Product_{1<=k<=n} (4sin(jPi/n)^2 + 4sin((2k-1)Pi/(2n))^2) ).

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

A340560 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-1} (4sin(aPi/n)^2 + 4sin(bPi/k)^2).

A340563 a(n) = sqrt( Product_{1<=j, k<=n-1} (4sin(jPi/n)^2 + 4cos(kPi/n)^2) ).