A253678 -id:A253678 - 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

A281583 Number of perfect matchings in the graph C_9 X C_{2n}.

Original entry on oeis.org

140450, 16091936, 2415542018, 400448833106, 69206906601800, 12190695635108354, 2167175327735637122, 387018647188487143424, 69272289588070930561250, 12413316310203106546620386, 2225719417041514241075539592, 399192630631160441128470998546

Offset: 2

Sergey Perepechko, Jan 25 2017

nonn

For odd values of m the order of recurrence relation for the number of perfect matchings in C_{m} X C_{2n} graph does not exceed 3^floor(m/2).

Seiichi Manyama, Table of n, a(n) for n = 2..443
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.
Sergey Perepechko, Generating function, in Maple notation.

Column k=9 of A341533.

Cf. A253678, A230033, A232804, A231485, A220864, A231087.

default(realprecision, 120);
a(n) = round(sqrt(prod(j=1, n, prod(k=1, 9, 4*sin((2*j-1)*Pi/(2*n))^2+4*sin((2*k-1)*Pi/9)^2)))); \\ Seiichi Manyama, Feb 14 2021

a(n) = sqrt( Product_{j=1..n} Product_{k=1..9} (4*sin((2*j-1)*Pi/(2*n))^2 + 4*sin((2*k-1)*Pi/9)^2) ). - Seiichi Manyama, Feb 14 2021

A281679 Number of perfect matchings in the graph C_10 X C_n.

Original entry on oeis.org

5054, 537636, 2540032, 114557000, 1034315998, 33898728836, 400448833106, 11203604497408, 152849502772958, 3876306765700644, 58099728840105682, 1375359477482867528, 22057225099289357824, 496348449090698237956, 8370856315868909044082, 181385918483215101487880

Offset: 3

Sergey Perepechko, Jan 26 2017

nonn

For even values of m the order of recurrence relation for the number of perfect matchings in C_m X C_n graph does not exceed (3^delta(m/2) + 2*(3/5)^(1 - delta(m/2)))*5^floor(m/4) + 1. Here delta(k) equals 1 for odd values of k and 0 otherwise. If m=10 the above estimate gives 126 for the order of recurrence relation while the exact value equals 118.

Seiichi Manyama, Table of n, a(n) for n = 3..500
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.
Sergey Perepechko, Generating function, in Maple notation.
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Matching
Eric Weisstein's World of Mathematics, Perfect Matching
Eric Weisstein's World of Mathematics, Torus Grid Graph

Row n=5 of A341741.

Cf. A220864, A230033, A231087, A231485, A232804, A253678, A281583.

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

Original entry on oeis.org

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

A308761 Number of perfect matchings in the graph C_{11} X C_{2n}.

Original entry on oeis.org

1956242, 643041038, 294554220578, 152849502772958, 83804387156528018, 47217865780262297342, 26990513247252188990402, 15550772782091243971206638, 8999393061535308152171682002, 5221063878050546380074377019392

Offset: 2

Sergey Perepechko, Jul 04 2019

nonn

This sequence satisfies a recurrence relation of order 243.

Seiichi Manyama, Table of n, a(n) for n = 2..361
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.
S. N. Perepechko, Counting Near-Perfect Matchings on C_m × C_n Tori of Odd Order in the Maple System, Programming and Computer Software, 45(2019), 65-72.
Sergey Perepechko, Generating function in Maple notation.

Column k=11 of A341533.

Cf. A230033, A231485, A232804, A253678, A281583, A281679.

default(realprecision, 120);
a(n) = round(sqrt(prod(j=1, n, prod(k=1, 11, 4*sin((2*j-1)*Pi/(2*n))^2+4*sin((2*k-1)*Pi/11)^2)))); \\ Seiichi Manyama, Feb 14 2021

a(n) = sqrt( Product_{j=1..n} Product_{k=1..11} (4*sin((2*j-1)*Pi/(2*n))^2 + 4*sin((2*k-1)*Pi/11)^2) ). - Seiichi Manyama, Feb 14 2021

A309018 Number of perfect matchings in the graph C_{12} X C_n.

Original entry on oeis.org

24200, 7379216, 41934482, 4357599552, 55820091938, 3827188349968, 69206906601800, 3876306765700644, 83804387156528018, 4161957566985310208, 100644292294423977842, 4601436044608986037284, 120511830300023778605000, 5179981855242249681088528, 144148769049390803580105218

Offset: 3

Sergey Perepechko, Jul 06 2019

nonn

This sequence satisfies a recurrence relation of order 266.

Seiichi Manyama, Table of n, a(n) for n = 3..500
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.
Sergey Perepechko, Generating function in Maple notation.

Row n=6 of A341741.

Cf. A230033, A231485, A232804, A253678, A281583, A281679, A308761.

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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A281583 Number of perfect matchings in the graph C_9 X C_{2n}.

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A281679 Number of perfect matchings in the graph C_10 X C_n.

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

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A308761 Number of perfect matchings in the graph C_{11} X C_{2n}.

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A309018 Number of perfect matchings in the graph C_{12} X C_n.

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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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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).