A281583 -id:A281583 - cp's OEIS frontend

A230033 Number of perfect matchings in the graph C_7 X C_{2n}.

10082, 401998, 19681538, 1034315998, 55820091938, 3044533460992, 166779871224962, 9152970837103102, 502711247500143362, 27619744381029252622, 1517688682641434229698, 83401213534557960429502, 4583249488240161816039552, 251871805990373105011941118, 13841645914590329223808310018, 760670944425011837491619633038

Offset: 2

Sergey Perepechko, Dec 20 2013

nonn

Seiichi Manyama, Table of n, a(n) for n = 2..500
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.

Column k=7 of A341533.

Cf. A231087, A220864, A231485, A232804, A281583, A308761.

default(realprecision, 120);
a(n) = round(sqrt(prod(j=1, n, prod(k=1, 7, 4*sin((2*j-1)*Pi/(2*n))^2+4*sin((2*k-1)*Pi/7)^2)))); \\ Seiichi Manyama, Feb 14 2021

G.f.: 2*x^2*(5041 -499700*x +20440353*x^2 -466963360*x^3 +6751799885*x^4 -66182756655*x^5 +459438362278*x^6 -2327864968019*x^7 +8797357131438*x^8 -25192378831195*x^9 +55291405473782*x^10 -93750343061691*x^11 +123440474579985*x^12 -126568817064424*x^13 +101127542456783*x^14 -62874205910076*x^15 +30308779015615*x^16 -11259345843608*x^17 +3194422598067*x^18 -683503915153*x^19 +108424368962*x^20 -12458825709*x^21 +1004282914*x^22 -54198917*x^23 +1818498*x^24 -33157*x^25 +239*x^26)/((1 -x)*(1 -13*x +57*x^2 -97*x^3 +57*x^4 -13*x^5 +x^6)*(1 -71*x +952*x^2 -3976*x^3 +6384*x^4 -3976*x^5 +952*x^6 -71*x^7 +x^8)*(1 -54*x +1039*x^2 -9096*x^3 +39037*x^4 -90378*x^5 +118951*x^6 -90378*x^7 +39037*x^8 -9096*x^9 +1039*x^10 -54*x^11 +x^12)).

a(n) = sqrt( Product_{j=1..n} Product_{k=1..7} (4*sin((2*j-1)*Pi/(2*n))^2 + 4*sin((2*k-1)*Pi/7)^2) ). - Seiichi Manyama, Feb 14 2021

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.

Original entry on oeis.org

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

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.

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

A230033 Number of perfect matchings in the graph C_7 X C_{2n}.

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

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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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Mathematica

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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).