A341741 -id:A341741 - cp's OEIS frontend

A103997 Square array T(M,N) read by antidiagonals: number of dimer tilings of a 2M X 2N Moebius strip.

1, 1, 1, 1, 3, 1, 1, 11, 7, 1, 1, 41, 71, 18, 1, 1, 153, 769, 539, 47, 1, 1, 571, 8449, 17753, 4271, 123, 1, 1, 2131, 93127, 603126, 434657, 34276, 322, 1, 1, 7953, 1027207, 20721019, 46069729, 10894561, 276119, 843, 1, 1, 29681, 11332097, 714790675, 4974089647, 3625549353, 275770321, 2226851, 2207, 1

Offset: 0

Ralf Stephan, Feb 26 2005

			Array begins:
  1,   1,     1,        1,          1,             1,               1,
  1,   3,     7,       18,         47,           123,             322,
  1,  11,    71,      539,       4271,         34276,          276119,
  1,  41,   769,    17753,     434657,      10894561,       275770321,
  1, 153,  8449,   603126,   46069729,    3625549353,    289625349454,
  1, 571, 93127, 20721019, 4974089647, 1234496016491, 312007855309063,
  ...

Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter and Tianyuan Xu, Sandpiles and Dominos, El. J. Comb., 22 (2015), P1.66. See Theorem 18.
W. T. Lu and F. Y. Wu, Dimer statistics on the Moebius strip and the Klein bottle, arXiv:cond-mat/9906154 [cond-mat.stat-mech], 1999.
Index entries for sequences related to dominoes

Rows include A005248, A103998.
Columns 1..7 give A001835(n+1), A334135, A334179, A334180, A334181, A334182, A334183.
Main diagonal gives A334124.
Cf. A099390, A103999, A341741, A334178, A256045, A348566.

T[M_, N_] := Product[4Sin[(4n-1)Pi/(4N)]^2 + 4Cos[m Pi/(2M+1)]^2, {n, 1, N}, {m, 1, M}];
Table[T[M - N, N] // Round, {M, 0, 9}, {N, 0, M}] // Flatten (* Jean-François Alcover, Dec 03 2018 *)

T(M, N) = Product_{m=1..M} (Product_{n=1..N} 4*sin(Pi*(4*n-1)/(4*N))^2 + 4*cos(Pi*m/(2*M + 1))^2).

For k > 0, T(n,k) = 2^n * sqrt(Resultant(U_{2*n}(x/2), T_{2*k}(i*x/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1). - Seiichi Manyama, Apr 15 2020

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)).

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)).

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.

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.

A341738 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = sqrt( Product_{a=1..n} Product_{b=1..k-1} (4sin((2a-1)Pi/(2n))^2 + 4sin(2b*Pi/k)^2) ).

Original entry on oeis.org

1, 2, 1, 7, 2, 1, 16, 25, 2, 1, 41, 72, 112, 2, 1, 98, 361, 400, 529, 2, 1, 239, 1250, 4961, 2312, 2527, 2, 1, 576, 5041, 25088, 77841, 13456, 12100, 2, 1, 1393, 18432, 200999, 559682, 1270016, 78408, 57967, 2, 1

Offset: 1

Seiichi Manyama, Feb 18 2021

			Square array begins:
  1, 2,     7,    16,       41,        98, ...
  1, 2,    25,    72,      361,      1250, ...
  1, 2,   112,   400,     4961,     25088, ...
  1, 2,   529,  2312,    77841,    559682, ...
  1, 2,  2527, 13456,  1270016,  12771458, ...
  1, 2, 12100, 78408, 20967241, 292820000, ...

Main diagonal gives A341782.

Cf. A341533, A341739, A341741.

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

If k is odd, T(n,k) = A341533(n,k)/2.

A341739 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} (4sin(aPi/n)^2 + 4sin((2b-1)Pi/(2k))^2).

Original entry on oeis.org

1, 1, 8, 1, 36, 49, 1, 200, 625, 288, 1, 1156, 12544, 9216, 1681, 1, 6728, 279841, 583200, 130321, 9800, 1, 39204, 6385729, 44408896, 24611521, 1822500, 57121, 1, 228488, 146410000, 3546167328, 6059221281, 1003520000, 25411681, 332928

Offset: 1

Seiichi Manyama, Feb 18 2021

			Square array begins:
     1,      1,        1,          1,             1, ...
     8,     36,      200,       1156,          6728, ...
    49,    625,    12544,     279841,       6385729, ...
   288,   9216,   583200,   44408896,    3546167328, ...
  1681, 130321, 24611521, 6059221281, 1612940640256, ...

Main diagonal gives A341478(n)^2.

Cf. A341533, A341738, A341741.

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

cp's OEIS Frontend

A103997 Square array T(M,N) read by antidiagonals: number of dimer tilings of a 2*M X 2*N Moebius strip.

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

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

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

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

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A341738 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = sqrt( Product_{a=1..n} Product_{b=1..k-1} (4*sin((2*a-1)*Pi/(2*n))^2 + 4*sin(2*b*Pi/k)^2) ).

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A341739 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} (4*sin(a*Pi/n)^2 + 4*sin((2*b-1)*Pi/(2*k))^2).

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A103997 Square array T(M,N) read by antidiagonals: number of dimer tilings of a 2M X 2N Moebius strip.

A341738 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = sqrt( Product_{a=1..n} Product_{b=1..k-1} (4sin((2a-1)Pi/(2n))^2 + 4sin(2b*Pi/k)^2) ).

A341739 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} (4sin(aPi/n)^2 + 4sin((2b-1)Pi/(2k))^2).