A220864 -id:A220864 - cp's OEIS frontend

A231485 Number of perfect matchings in the graph C_5 X C_{2n}.

722, 9922, 155682, 2540032, 41934482, 694861522, 11527389122, 191304901282, 3175220160032, 52703408458882, 874800747092322, 14520494659638322, 241020661471736882, 4000620276282860032, 66404949893677073282, 1102233473331064193122, 18295603728585969257522

Offset: 2

Sergey Perepechko, Nov 09 2013

Colin Barker, Table of n, a(n) for n = 2..819
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.
Index entries for linear recurrences with constant coefficients, signature (29,-261,1029,-2001,2001,-1029,261,-29,1).

Cf. A220864, A231087.

CoefficientList[Series[2*(361 - 5508*x + 28193*x^2 - 64021*x^3 + 70770*x^4 - 38841*x^5 + 10278*x^6 - 1173*x^7 + 41*x^8)/((1 - x)*(1 - 9*x + 21*x^2 - 9*x^3 + x^4)*(1 - 19*x + 41*x^2 - 19*x^3 + x^4)), {x, 0, 20}], x] (* Wesley Ivan Hurt, Jul 17 2025 *)

Vec(2*x^2*(361-5508*x+28193*x^2-64021*x^3+70770*x^4-38841*x^5+10278*x^6-1173*x^7+41*x^8)/((1-x)*(1-9*x+21*x^2-9*x^3+x^4)*(1-19*x+41*x^2-19*x^3+x^4)) + O(x^100)) \\ Colin Barker, Dec 13 2014

default(realprecision, 120);
a(n) = round(sqrt(prod(j=1, n, prod(k=1, 5, 4*sin((2*j-1)*Pi/(2*n))^2+4*sin((2*k-1)*Pi/5)^2)))); \\ Seiichi Manyama, Feb 14 2021

G.f.: 2*x^2*(361-5508*x+28193*x^2-64021*x^3+70770*x^4-38841*x^5+10278*x^6-1173*x^7+41*x^8)/((1-x)*(1-9*x+21*x^2-9*x^3+x^4)*(1-19*x+41*x^2-19*x^3+x^4)).
From Seiichi Manyama, Feb 14 2021: (Start)
a(n) = sqrt( Product_{j=1..n} Product_{k=1..5} (4*sin((2*j-1)*Pi/(2*n))^2 + 4*sin((2*k-1)*Pi/5)^2) ).
a(n) = 28*a(n-1) - 233*a(n-2) + 796*a(n-3) - 1205*a(n-4) + 796*a(n-5) - 233*a(n-6) + 28*a(n-7) - a(n-8) + 200. (End)

A231087 Number of perfect matchings in graph C_3 x C_{2n}.

Original entry on oeis.org

50, 224, 1058, 5054, 24200, 115934, 555458, 2661344, 12751250, 61094894, 292723208, 1402521134, 6719882450, 32196891104, 154264573058, 739125974174, 3541365297800, 16967700514814, 81297137276258, 389517985866464, 1866292792056050, 8941945974413774, 42843437080012808, 205275239425650254

Offset: 2

Sergey Perepechko, Nov 03 2013

Index entries for sequences related to dominoes
Index entries for linear recurrences with constant coefficients, signature (6, -6, 1)

Cf. A054493, A220864.

Vec(2*x^2*(25-38*x+7*x^2)/((1-x)*(1-5*x+x^2))+O(x^66)) \\ Joerg Arndt, Nov 03 2013

default(realprecision, 120);
a(n) = round(sqrt(prod(j=1, n, prod(k=1, 3, 4*sin((2*j-1)*Pi/(2*n))^2+4*sin((2*k-1)*Pi/3)^2)))); \\ Seiichi Manyama, Feb 14 2021

a(n) = 2*(((sqrt(7)+sqrt(3))/2)^n + ((sqrt(7)-sqrt(3))/2)^n)^2.
G.f.: 2*x^2*(25-38*x+7*x^2)/((1-x)*(1-5*x+x^2)).
From Seiichi Manyama, Feb 14 2021: (Start)
a(n) = sqrt( Product_{j=1..n} Product_{k=1..3} (4*sin((2*j-1)*Pi/(2*n))^2 + 4*sin((2*k-1)*Pi/3)^2) ).
a(n) = 5*a(n-1) - a(n-2) - 12. (End)
a(n) = 6*A054493(n-1) + 8. - Peter Bala, May 17 2025

A232804 Number of perfect matchings in the graph C_6 x C_n.

Original entry on oeis.org

224, 3108, 9922, 90176, 401998, 3113860, 16091936, 114557000, 643041038, 4357599552, 25689719122, 169094614280, 1026275640544, 6640849944580, 40998347400722, 262671237617216, 1637828186763038, 10433179552323108, 65428999765032736, 415409841636546440, 2613799160004664798, 16563343174199239744

Offset: 3

Sergey Perepechko, Nov 30 2013

nonn

Seiichi Manyama, Table of n, a(n) for n = 3..500
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.

Row n=3 of A341741.

Cf. A220864, A231087, A231485.

G.f: 2*x^3*(112+882*x-8955*x^2-22184*x^3+151298*x^4+192108*x^5-1004174*x^6-773678*x^7+3077791*x^8+1598624*x^9-4646368*x^10-1738444*x^11+3589216*x^12+ 1010882*x^13-1408253*x^14-318388*x^15+271982*x^16+52648*x^17-23250*x^18-4062*x^19+601*x^20+100*x^21)/((1-x)*(1+x)*(1+5*x+x^2)*(1-5*x+x^2)*(1-2*x-x^2)* (1+2*x-x^2)*(1+x-x^2)*(1-x-x^2)*(1-5*x^2+x^4)*(1-6*x-3*x^2+6*x^3+x^4)).

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

Original entry on oeis.org

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

A253678 Number of perfect matchings in the graph C_8 X C_n.

Original entry on oeis.org

1058, 39952, 155682, 3113860, 19681538, 311853312, 2415542018, 33898728836, 294554220578, 3827188349968, 35866638601250, 442299574618756, 4365923647238658, 51942700201804032, 531410627302657538, 6169093269471927940, 64681086501382749218, 738453913359765339152, 7872683691901209561122, 88873260229652630182276

Offset: 3

Sergey Perepechko, Jan 09 2015

nonn

S. N. Perepechko, Combinatorial properties of dimer problem on tori (in Russian). Mathematical physics and its applications, The fourth int. conf. Samara, 2014, 280-281.

Seiichi Manyama, Table of n, a(n) for n = 3..500
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=4 of A341741.

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

a(n) = 14*a(n-1) + 145*a(n-2) - 2492*a(n-3) - 5832*a(n-4) + 164332*a(n-5) + 6360*a(n-6) - 5592188*a(n-7) + 5575094*a(n-8) + 111829704*a(n-9) - 176471286*a(n-10) - 1404071060*a(n-11) + 2757391176*a(n-12) + 11493707876*a(n-13) - 26094214040*a(n-14) - 62666476628*a(n-15) + 161092194209*a(n-16) + 229194775110*a(n-17) - 673504262865*a(n-18) - 556186915928*a(n-19) + 1946775340976*a(n-20) + 855365272888*a(n-21) - 3933950269712*a(n-22) - 705783359960*a(n-23) + 5586898052980*a(n-24) - 5586898052980*a(n-26) + 705783359960*a(n-27) + 3933950269712*a(n-28) - 855365272888*a(n-29) - 1946775340976*a(n-30) + 556186915928*a(n-31) + 673504262865*a(n-32) - 229194775110*a(n-33) - 161092194209*a(n-34) + 62666476628*a(n-35) + 26094214040*a(n-36) - 11493707876*a(n-37) - 2757391176*a(n-38) + 1404071060*a(n-39) + 176471286*a(n-40) - 111829704*a(n-41) - 5575094*a(n-42) + 5592188*a(n-43) - 6360*a(n-44) - 164332*a(n-45) + 5832*a(n-46) + 2492*a(n-47) - 145*a(n-48) - 14*a(n-49) + a(n-50).

G.f.: 2*x^3*(529 + 12570*x - 278528*x^2 - 1111096*x^3 + 29622124*x^4 + 15949216*x^5 - 1354335880*x^6 + 1073870160*x^7 + 33231636934*x^8 - 49093408612*x^9 - 484852497568*x^10 + 922702092728*x^11 + 4448623050276*x^12 - 9889298009728*x^13 - 26519860399096*x^14 + 66909591407824*x^15 + 104242913448099*x^16 - 300153880511538*x^17 - 268804327853184*x^18 + 917127529551440*x^19 + 437177534552376*x^20 - 1937370697752896*x^21 - 386856893695952*x^22 + 2851262465341600*x^23 + 31463729114724*x^24 - 2933939639544920*x^25 + 353114911609152*x^26 + 2113468417316080*x^27 - 452714140134072*x^28 - 1064902306141568*x^29 + 302352881352848*x^30 + 373692292484128*x^31 - 126783009087417*x^32 - 90391126093930*x^33 + 35100066280832*x^34 + 14772327002472*x^35 - 6497628908516*x^36 - 1572040067936*x^37 + 799287715544*x^38 + 101192826896*x^39 - 63992712074*x^40 - 3215530756*x^41 + 3212411488*x^42 - 3162664*x^43 - 94666796*x^44 + 3355392*x^45 + 1438440*x^46 - 83696*x^47 - 8091*x^48 + 578*x^49)/((1-x)*(1+x)*(1+4*x+x^2)*(1-4*x+x^2)*(1-2*x-x^2)*(1+2*x-x^2)*(1+8*x+16*x^2+8*x^3+x^4)* (1-14*x+34*x^2-14*x^3+x^4)*(1-8*x+16*x^2-8*x^3+x^4)*(1-4*x^2+x^4)*(1+4*x-4*x^2-4*x^3+x^4)*(1+8*x-10*x^2-8*x^3+x^4)*(1-4*x-4*x^2+4*x^3+x^4)*(1-8*x-10*x^2+8*x^3+x^4)*(1-14*x^2+34*x^4-14*x^6+x^8)).

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

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

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

A231485 Number of perfect matchings in the graph C_5 X C_{2n}.

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A231087 Number of perfect matchings in graph C_3 x C_{2n}.

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A232804 Number of perfect matchings in the graph C_6 x C_n.

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

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A253678 Number of perfect matchings in the graph C_8 X C_n.

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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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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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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).