A341533 -id:A341533 - 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

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

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

A341535 a(n) = sqrt(Product_{1<=j,k<=n} (4sin((2j-1)Pi/(2n))^2 + 4sin((2k-1)*Pi/n)^2)).

Original entry on oeis.org

1, 2, 36, 224, 38416, 2540032, 4115479104, 3044533460992, 48656376372265216, 387018647188487143424, 62441634466575620320306176, 5221063878050546380074377019392, 8590392749565593082105293619707908096, 7476351474500749779460880888573410601336832

Offset: 0

Seiichi Manyama, Feb 13 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..62

Main diagonal of A341533.

Cf. A341478, A341479, A341782.

Table[Sqrt[Product[4*Sin[(2*j - 1)*Pi/(2*n)]^2 + 4*Sin[(2*k - 1)*Pi/n]^2, {k, 1, n}, {j, 1, n}]], {n, 0, 20}] // Round (* Vaclav Kotesovec, Feb 14 2021 *)

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

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

If n is odd, a(n) = 2*A341478(n). - Seiichi Manyama, Feb 19 2021

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)));

A341543 a(n) = sqrt( Product_{j=1..n} Product_{k=1..2} (4sin((2j-1)Pi/(2n))^2 + 4sin((2k-1)*Pi/2)^2) ).

Original entry on oeis.org

8, 36, 200, 1156, 6728, 39204, 228488, 1331716, 7761800, 45239076, 263672648, 1536796804, 8957108168, 52205852196, 304278005000, 1773462177796, 10336495061768, 60245508192804, 351136554095048, 2046573816377476, 11928306344169800

Offset: 1

Seiichi Manyama, Feb 14 2021

nonn

Index entries for linear recurrences with constant coefficients, signature (7, -7, 1)

Column k=2 of A341533.

Cf. A001541.

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

a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3).
a(n) = 6*a(n-1) - a(n-2) - 8.
a(n) = 2*(A001541(n) + 1). - Hugo Pfoertner, Feb 14 2021
G.f.: 4*x*(2 - 5*x + x^2)/((1 - x)*(1 - 6*x + x^2)). - Vaclav Kotesovec, Feb 14 2021

A341544 a(n) = sqrt( Product_{j=1..n} Product_{k=1..4} (4sin((2j-1)Pi/(2n))^2 + 4sin((2k-1)*Pi/4)^2) ).

Original entry on oeis.org

36, 256, 2916, 38416, 527076, 7311616, 101727396, 1416468496, 19727326116, 274760478976, 3826898412516, 53301739046416, 742397156205156, 10340257357947136, 144021201787572516, 2005956552488017936, 27939370476391960356, 389145229905568604416, 5420093847412497929316

Offset: 1

Seiichi Manyama, Feb 14 2021

nonn

Index entries for linear recurrences with constant coefficients, signature (19, -76, 76, -19, 1)

Column k=4 of A341533.

Table[6 + 4 (2 + Sqrt[3])^n + 4 (2 - Sqrt[3])^n + (7 + 4 Sqrt[3])^n + (7 - 4 Sqrt[3])^n, {n, 1, 20}] // FullSimplify (* Vaclav Kotesovec, Feb 14 2021 *)

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

a(n) = 19*a(n-1) - 76*a(n-2) + 76*a(n-3) - 19*a(n-4) + a(n-5).
a(n) = 18*a(n-1) - 58*a(n-2) + 18*a(n-3) - a(n-4) + 144.
From Vaclav Kotesovec, Feb 14 2021: (Start)
G.f.: 4*(4 - 67*x + 197*x^2 - 107*x^3 + 9*x^4) / ((1 - x)*(1 - 14*x + x^2)*(1 - 4*x + x^2)).
a(n) = 6 + 4*(2 + sqrt(3))^n + 4*(2 - sqrt(3))^n + (7 + 4*sqrt(3))^n + (7 - 4*sqrt(3))^n. (End)

A341545 a(n) = sqrt( Product_{j=1..n} Product_{k=1..6} (4sin((2j-1)Pi/(2n))^2 + 4sin((2k-1)*Pi/6)^2) ).

Original entry on oeis.org

200, 2916, 80000, 2775556, 105125000, 4115479104, 163146144200, 6498349262596, 259309319120000, 10354620147583716, 413585320648104200, 16521137110112348224, 659981119616472888200, 26365103950427540487396, 1053246219256801250000000

Offset: 1

Seiichi Manyama, Feb 14 2021

nonn

Index entries for linear recurrences with constant coefficients, signature (77, -2002, 24596, -165165, 653835, -1598883, 2483481, -2483481, 1598883, -653835, 165165, -24596, 2002, -77, 1)

Column k=6 of A341533.

default(realprecision, 120);
a(n) = round(sqrt(prod(j=1, n, prod(k=1, 6, 4*sin((2*j-1)*Pi/(2*n))^2+4*sin((2*k-1)*Pi/6)^2))));

a(n) = 77*a(n-1) - 2002*a(n-2) + 24596*a(n-3) - 165165*a(n-4) + 653835*a(n-5) - 1598883*a(n-6) + 2483481*a(n-7) - 2483481*a(n-8) + 1598883*a(n-9) - 653835*a(n-10) + 165165*a(n-11) - 24596*a(n-12) + 2002*a(n-13) - 77*a(n-14) + a(n-15).
a(n) = 76*a(n-1) - 1926*a(n-2) + 22670*a(n-3) - 142495*a(n-4) + 511340*a(n-5) - 1087543*a(n-6) + 1395938*a(n-7) - 1087543*a(n-8) + 511340*a(n-9) - 142495*a(n-10) + 22670*a(n-11) - 1926*a(n-12) + 76*a(n-13) - a(n-14) - 2160.
G.f.: 4*(50*x - 3121*x^2 + 63967*x^3 - 616453*x^4 + 3219563*x^5 - 9827161*x^6 + 18330389*x^7 - 21405307*x^8 + 15754967*x^9 - 7241797*x^10 + 2026187*x^11 - 329569*x^12 + 28911*x^13 - 1182*x^14 + 16*x^15) / ((1 - x)*(1 - 7*x + x^2)*(1 - 6*x + x^2)*(1 - 3*x + x^2)* (1 - 42*x + 83*x^2 - 42*x^3 + x^4)*(1 - 18*x + 43*x^2 - 18*x^3 + x^4)). - 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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A281583 Number of perfect matchings in the graph C_9 X C_{2n}.

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

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A341535 a(n) = sqrt(Product_{1<=j,k<=n} (4*sin((2*j-1)*Pi/(2*n))^2 + 4*sin((2*k-1)*Pi/n)^2)).

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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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A341543 a(n) = sqrt( Product_{j=1..n} Product_{k=1..2} (4*sin((2*j-1)*Pi/(2*n))^2 + 4*sin((2*k-1)*Pi/2)^2) ).

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A341544 a(n) = sqrt( Product_{j=1..n} Product_{k=1..4} (4*sin((2*j-1)*Pi/(2*n))^2 + 4*sin((2*k-1)*Pi/4)^2) ).

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A341535 a(n) = sqrt(Product_{1<=j,k<=n} (4sin((2j-1)Pi/(2n))^2 + 4sin((2k-1)*Pi/n)^2)).

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).

A341543 a(n) = sqrt( Product_{j=1..n} Product_{k=1..2} (4sin((2j-1)Pi/(2n))^2 + 4sin((2k-1)*Pi/2)^2) ).

A341544 a(n) = sqrt( Product_{j=1..n} Product_{k=1..4} (4sin((2j-1)Pi/(2n))^2 + 4sin((2k-1)*Pi/4)^2) ).

A341545 a(n) = sqrt( Product_{j=1..n} Product_{k=1..6} (4sin((2j-1)Pi/(2n))^2 + 4sin((2k-1)*Pi/6)^2) ).