A335586 -id:A335586 - cp's OEIS frontend

A341741 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards: number of perfect matchings in the graph C_{2n} x C_k.

2, 8, 2, 14, 36, 2, 36, 50, 200, 2, 82, 272, 224, 1156, 2, 200, 722, 3108, 1058, 6728, 2, 478, 3108, 9922, 39952, 5054, 39204, 2, 1156, 10082, 90176, 155682, 537636, 24200, 228488, 2, 2786, 39952, 401998, 3113860, 2540032, 7379216, 115934, 1331716, 2

Offset: 1

Seiichi Manyama, Feb 18 2021

Dimer tilings of 2n x k toroidal grid.

			Square array begins:
  2,     8,    14,      36,       82,        200, ...
  2,    36,    50,     272,      722,       3108, ...
  2,   200,   224,    3108,     9922,      90176, ...
  2,  1156,  1058,   39952,   155682,    3113860, ...
  2,  6728,  5054,  537636,  2540032,  114557000, ...
  2, 39204, 24200, 7379216, 41934482, 4357599552, ...

Seiichi Manyama, Antidiagonals n = 1..50, flattened
P. W. Kasteleyn, The statistics of dimers on a lattice, I. the number of dimer arrangements on a quadratic lattice, Physica 27 (1961), 1209-1225. See Eq. (25).
Index entries for sequences related to dominoes

Columns 1..12 give A007395, A162484(2*n), A231087, A220864(2*n), A231485, A232804(2*n), A230033, A253678(2*n), A281583, A281679(2*n), A308761, A309018(2*n).
Rows 1..6 give A162484, A220864, A232804, A253678, A281679, A309018.
T(n,2*n) gives A335586.
Cf. A341533, A341738, A341739.
Cf. A099390, A103997, A103999.

T(n,k) = A341533(n,k)/2 + A341738(n,k) + 2 * ((k+1) mod 2) * A341739(n,ceiling(k/2)).
T(n, 2k) = T(k, 2n).
If k is odd, T(n,k) = A341533(n,k) = 2*A341738(n,k).

New name from Andrey Zabolotskiy, Dec 26 2021

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

A341493 a(n) = ( Product_{j=1..n} Product_{k=1..n+1} (4sin((2j-1)Pi/n)^2 + 4sin((2k-1)Pi/(n+1))^2) )^(1/4).

Original entry on oeis.org

1, 2, 14, 50, 722, 9922, 401998, 19681538, 2415542018, 400448833106, 152849502772958, 83804387156528018, 100644292294423977842, 180483873668860889130642, 686161117968330536875295134, 4001215836806010384390623471618

Offset: 0

Seiichi Manyama, Feb 13 2021

nonn

Number of perfect matchings in the graph C_n X C_{n+1} for n > 0.

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.
Eric Weisstein's World of Mathematics, Perfect Matching
Eric Weisstein's World of Mathematics, Torus Grid Graph

Cf. A162484, A220864, A230033, A231087, A231485, A232804, A253678, A281583, A281679, A308761, A309018, A335586.

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

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

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

cp's OEIS Frontend

A341741 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards: number of perfect matchings in the graph C_{2n} x C_k.

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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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A341493 a(n) = ( Product_{j=1..n} Product_{k=1..n+1} (4*sin((2*j-1)*Pi/n)^2 + 4*sin((2*k-1)*Pi/(n+1))^2) )^(1/4).

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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).

A341493 a(n) = ( Product_{j=1..n} Product_{k=1..n+1} (4sin((2j-1)Pi/n)^2 + 4sin((2k-1)Pi/(n+1))^2) )^(1/4).