A099390 -id:A099390 - cp's OEIS frontend

A065072 Number of ways to tile a square of side 2n by dominoes (rectangles of size 2 X 1 or 1 X 2) is 2^n * a(n)^2 (see A004003).

1, 1, 3, 29, 901, 89893, 28793575, 29607089625, 97725875584681, 1035449388414303593, 35216739783694029601963, 3844747107219467355553841461, 1347358497824862447450096142795629, 1515633798331963142551890627742773295309

Offset: 0

Nicolau C. Saldanha (nicolau(AT)mat.puc-rio.br), Nov 08 2001

nonn

A099390 is the main entry for this problem. - N. J. A. Sloane, Mar 15 2015

			G.f. = 1 + x + 3*x^2 + 29*x^3 + 901*x^4 + 89893*x^5 + 28793575*x^6 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..50 (terms n=1..25 from T. D. Noe)
N. Allegra, Exact solution of the 2d dimer model: Corner free energy, correlation functions and combinatorics, arXiv:1410.4131 [cond-mat.stat-mech], 2014.
H. Cohn, 2-adic behavior of numbers of domino tilings, arXiv:math/0008222 [math.CO], 2000.
Philippe Di Francesco, Twenty Vertex model and domino tilings of the Aztec triangle, arXiv:2102.02920 [math.CO], 2021. Mentions this sequence.
Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter, and Tianyuan Xu, Sandpiles and Dominos, Electronic Journal of Combinatorics, Volume 22(1), 2015. [Mentions this sequence together with a different sequence (A256043) with the same initial terms]
Peter E. John and Horst Sachs, On a strange observation in the theory of the dimer problem, Discrete Math. 216 (2000), no. 1-3, 211-219. [_N. J. A. Sloane_, Feb 06 2012]
James Propp, Some 2-Adic Conjectures Concerning Polyomino Tilings of Aztec Diamonds, Integers (2023) Vol. 23, Art. A30.

Cf. A099390, A004003, A256043.

a[n_] := With[{L = 2n}, Sqrt[Product[4 Cos[p Pi/(L+1)]^2 + 4 Cos[q Pi/(L+1)]^2, {p, 1, L/2}, {q, 1, L/2}]/2^(L/2)] // Round];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Nov 11 2018 *)
Table[Resultant[ChebyshevU[2*n, x/2], ChebyshevU[2*n, I*x/2], x]^(1/4) / 2^(n/2), {n, 0, 15}] (* Vaclav Kotesovec, Dec 30 2020 *)

a(n) ~ exp(G*(2*n + 1)^2/(2*Pi)) / (2^((n-1)/2) * (1 + sqrt(2))^(n + 1/2)), where G is Catalan's constant A006752. - Vaclav Kotesovec, Apr 14 2020, updated Dec 30 2020

a(0)=1 prepended by Alois P. Heinz, Mar 25 2015

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

Original entry on oeis.org

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

A187596 Array T(m,n) read by antidiagonals: number of domino tilings of the m X n grid (m>=0, n>=0).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 3, 3, 1, 1, 1, 0, 5, 0, 5, 0, 1, 1, 1, 8, 11, 11, 8, 1, 1, 1, 0, 13, 0, 36, 0, 13, 0, 1, 1, 1, 21, 41, 95, 95, 41, 21, 1, 1, 1, 0, 34, 0, 281, 0, 281, 0, 34, 0, 1, 1, 1, 55, 153, 781, 1183, 1183, 781, 153, 55, 1, 1, 1, 0, 89, 0, 2245, 0, 6728, 0, 2245, 0, 89, 0, 1, 1, 1, 144, 571, 6336

Offset: 0

N. J. A. Sloane, Mar 11 2011

A099390 supplemented by an initial row and column of 1's.
See A099390 (the main entry for this array) for further information.
If we work with the row index starting at 1 then every row of the array is a divisibility sequence, i.e., the terms satisfy the property that if n divides m then a(n) divide a(m) provided a(n) != 0. Row k satisfies a linear recurrence of order 2^floor(k/2) (Stanley, Ex. 36 p. 273). - Peter Bala, Apr 30 2014

			Array begins:
  1,  1,  1,  1,   1,    1,     1,     1,       1,      1,        1, ...
  1,  0,  1,  0,   1,    0,     1,     0,       1,      0,        1, ...
  1,  1,  2,  3,   5,    8,    13,    21,      34,     55,       89, ...
  1,  0,  3,  0,  11,    0,    41,     0,     153,      0,      571, ...
  1,  1,  5, 11,  36,   95,   281,   781,    2245,   6336,    18061, ...
  1,  0,  8,  0,  95,    0,  1183,     0,   14824,      0,   185921, ...
  1,  1, 13, 41, 281, 1183,  6728, 31529,  167089, 817991,  4213133, ...
  1,  0, 21,  0, 781,    0, 31529,     0, 1292697,      0, 53175517, ...

R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, 1997.

Alois P. Heinz, Antidiagonals n = 0..80, flattened
James Propp, Enumeration of Matchings: Problems and Progress, arXiv:math/9904150 [math.CO], 1999.
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the second kind.
Eric Weisstein's World of Mathematics, Fibonacci Polynomial.

Cf. A099390.
See A187616 for a triangular version, and A187617, A187618 for the sub-array T(2m,2n).
See also A049310, A053117.

with(LinearAlgebra):
T:= proc(m,n) option remember; local i, j, t, M;
      if m<=1 or n<=1 then 1 -irem(n*m, 2)
    elif irem(n*m, 2)=1 then 0
    elif mAlois P. Heinz, Apr 11 2011

t[m_, n_] := Product[2*(2+Cos[2*j*Pi/(m+1)]+Cos[2*k*Pi/(n+1)]), {k, 1, n/2}, {j, 1, m/2}]; t[?OddQ, ?OddQ] = 0; Table[t[m-n, n] // FullSimplify, {m, 0, 13}, {n, 0, m}] // Flatten (* Jean-François Alcover, Jan 07 2014, after A099390 *)

From Peter Bala, Apr 30 2014: (Start)
T(n,k)^2 = absolute value of Product_{b=1..k} Product_{a=1..n} ( 2*cos(a*Pi/(n+1)) + 2*i*cos(b*Pi/(k+1)) ), where i = sqrt(-1). See Propp, Section 5.
Equivalently, working with both the row index n and column index k starting at 1 we have T(n,k)^2 = absolute value of Resultant (F(n,x), U(k-1,x/2)), where U(n,x) is a Chebyshev polynomial of the second kind and F(n,x) is a Fibonacci polynomial defined recursively by F(0,x) = 0, F(1,x) = 1 and F(n,x) = x*F(n-1,x) + F(n-2,x) for n >= 2. The divisibility properties of the array entries mentioned in the Comments are a consequence of this result. (End)

A239264 Number A(n,k) of domicule tilings of a k X n grid; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 3, 0, 1, 1, 1, 5, 5, 1, 1, 1, 0, 11, 0, 11, 0, 1, 1, 1, 21, 43, 43, 21, 1, 1, 1, 0, 43, 0, 280, 0, 43, 0, 1, 1, 1, 85, 451, 1563, 1563, 451, 85, 1, 1, 1, 0, 171, 0, 9415, 0, 9415, 0, 171, 0, 1, 1, 1, 341, 4945, 55553, 162409, 162409, 55553, 4945, 341, 1, 1

Offset: 0

Alois P. Heinz, Mar 13 2014

A domicule is either a domino or it is formed by the union of two neighboring unit squares connected via their corners. In a tiling the connections of two domicules are allowed to cross each other.

			A(3,2) = 5:
  +-----+ +-----+ +-----+ +-----+ +-----+
  |o o-o| |o o o| |o o o| |o o o| |o-o o|
  ||    | ||  X | || | || | X  || |    ||
  |o o-o| |o o o| |o o o| |o o o| |o-o o|
  +-----+ +-----+ +-----+ +-----+ +-----+
A(4,3) = 43:
  +-------+ +-------+ +-------+ +-------+ +-------+
  |o o o o| |o o o-o| |o o-o o| |o o-o o| |o o-o o|
  ||  X  || | X     | | \   / | ||     || | \    ||
  |o o o o| |o o o o| |o o o o| |o o o o| |o o o o|
  |       | |     X | ||     || |   \ \ | ||    \ |
  |o-o o-o| |o-o o o| |o o-o o| |o-o o o| |o o-o o|
  +-------+ +-------+ +-------+ +-------+ +-------+ ...
Square array A(n,k) begins:
  1, 1,  1,   1,    1,      1,       1, ...
  1, 0,  1,   0,    1,      0,       1, ...
  1, 1,  3,   5,   11,     21,      43, ...
  1, 0,  5,   0,   43,      0,     451, ...
  1, 1, 11,  43,  280,   1563,    9415, ...
  1, 0, 21,   0, 1563,      0,  162409, ...
  1, 1, 43, 451, 9415, 162409, 3037561, ...

Alois P. Heinz, Antidiagonals n = 0..36, flattened

Columns (or rows) k=0-10 give: A000012, A059841, A001045(n+1), A239265, A239266, A239267, A239268, A239269, A239270, A239271, A239272.
Bisection of main diagonal gives: A239273.
Cf. A099390, A187616, A187617, A187596, A220644.

b:= proc(n, l) option remember; local d, f, k;
      d:= nops(l)/2; f:=false;
      if n=0 then 1
    elif l[1..d]=[f$d] then b(n-1, [l[d+1..2*d][], true$d])
    else for k to d while not l[k] do od;
         `if`(k1 and l[k+d+1],
                              b(n, subsop(k=f, k+d+1=f, l)), 0)+
         `if`(k>1 and n>1 and l[k+d-1],
                              b(n, subsop(k=f, k+d-1=f, l)), 0)+
         `if`(n>1 and l[k+d], b(n, subsop(k=f, k+d=f, l)), 0)+
         `if`(k `if`(irem(n*k, 2)>0, 0,
    `if`(k>n, A(k, n), b(n, [true$(k*2)]))):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, l_List] := b[n, l] = Module[{d = Length[l]/2, f = False, k}, Which [n == 0, 1, l[[1 ;; d]] == Array[f&, d], b[n-1, Join[l[[d+1 ;; 2*d]], Array[True&, d]]], True, For[k=1, !l[[k]], k++]; If[k1 && l[[k+d+1]], b[n, ReplacePart[l, {k -> f, k+d+1 -> f}]], 0] + If[k>1 && n>1 && l[[k+d-1]], b[n, ReplacePart[l, {k -> f, k+d-1 -> f}]], 0] + If[n>1 && l[[k+d]], b[n, ReplacePart[l, {k -> f, k+d -> f}]], 0] + If[k f, k+1 -> f}]], 0]]]; A[n_, k_] := If[Mod[n*k, 2]>0, 0, If[k>n, A[k, n], b[n, Array[True&, k*2]]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Feb 02 2015, after Alois P. Heinz *)

A187617 Array T(m,n) read by antidiagonals: number of domino tilings of the 2m X 2n grid (m>=0, n>=0).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 13, 36, 13, 1, 1, 34, 281, 281, 34, 1, 1, 89, 2245, 6728, 2245, 89, 1, 1, 233, 18061, 167089, 167089, 18061, 233, 1, 1, 610, 145601, 4213133, 12988816, 4213133, 145601, 610, 1

Offset: 0

N. J. A. Sloane, Mar 11 2011

A099390 is the main entry for this problem.
The even-indexed rows and columns of the square array in A187596.
Row (and column) 2 is given by A122367. - Nathaniel Johnston, Mar 22 2011

			The array begins:
  1,  1,     1,       1,          1,            1, ...
  1,  2,     5,      13,         34,           89, ...
  1,  5,    36,     281,       2245,        18061, ...
  1, 13,   281,    6728,     167089,      4213133, ...
  1, 34,  2245,  167089,   12988816,   1031151241, ...
  1, 89, 18061, 4213133, 1031151241, 258584046368, ...

Alois P. Heinz, Antidiagonals n = 0..26, flattened
N. Allegra, Exact solution of the 2d dimer model: Corner free energy, correlation functions and combinatorics, arXiv:1410.4131 [cond-mat.stat-mech], 2014. See Table 1.
Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter and Tianyuan Xu, Sandpiles and Dominos, El. J. Comb., 22 (2015), P1.66. See Theorem 15.
Index entries for sequences related to dominoes

A187618 is the triangle version.
Cf. A187596, A099390, A348566.
Main diagonal is A004003. Second and third rows give A001519, A188899.

ft:=(m,n)->
2^(m*n/2)*mul( mul(
(cos(Pi*i/(n+1))^2+cos(Pi*j/(m+1))^2), j=1..m/2), i=1..n/2);
T:=(m,n)->round(evalf(ft(m,n),300));

T[m_, n_] := Product[2(2 + Cos[(2j Pi)/(2m+1)] + Cos[(2k Pi)/(2n+1)]), {j, 1, m}, {k, 1, n}];
Table[T[m-n, n] // Round, {m, 0, 8}, {n, 0, m}] // Flatten (* Jean-François Alcover, Aug 05 2018 *)

default(realprecision, 120);
{T(n, k) = round(prod(a=1, n, prod(b=1, k, 4*cos(a*Pi/(2*n+1))^2+4*cos(b*Pi/(2*k+1))^2)))} \\ Seiichi Manyama, Jan 09 2021

More terms from Nathaniel Johnston, Mar 22 2011

A003775 Number of perfect matchings (or domino tilings) in P_5 X P_2n.

Original entry on oeis.org

1, 8, 95, 1183, 14824, 185921, 2332097, 29253160, 366944287, 4602858719, 57737128904, 724240365697, 9084693297025, 113956161827912, 1429438110270431, 17930520634652959, 224916047725262248, 2821291671062267585, 35389589910135145793, 443918325373278904936

Offset: 0

Frans J. Faase

nonn

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.
R. P. Stanley, Enumerative Combinatorics I, p. 292.

Alois P. Heinz, Table of n, a(n) for n = 0..300
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
David Klarner and Jordan Pollack, Domino tilings of rectangles with fixed width, Disc. Math. 32 (1980) 45-52.
R. J. Mathar, Paving rectangular regions with rectangular tiles: tatami and non-tatami tilings, arXiv:1311.6135 [math.CO], 2013, Table 4.
James A. Sellers, Domino Tilings and Products of Fibonacci and Pell Numbers, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.2.
Thotsaporn "Aek" Thanatipanonda, Statistics of Domino Tilings on a Rectangular Board, Fibonacci Quart. 57 (2019), no. 5, 145-153. See p. 151.
Index entries for sequences related to dominoes
Index entries for linear recurrences with constant coefficients, signature (15,-32,15,-1).

Row 5 of array A099390.

Bisection of A189003.

I:=[1,8,95,1183]; [n le 4 select I[n] else 15*Self(n-1)-32*Self(n-2)+15*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Aug 20 2018

a:= n-> (<<15|-32|15|-1>, <1|0|0|0>, <0|1|0|0>, <0|0|1|0>>^n. <<8, 1, 1, 8>>)[2, 1]: seq(a(n), n=0..20);  # Alois P. Heinz, Sep 24 2011

a = 3; b = 5; c = 7; d = a*c; e = b*c; g = a*b*c; f[n_] := Simplify[((e + c*Sqrt[b] + b*Sqrt[d] + Sqrt[g])*((a + Sqrt[b])*(b + Sqrt[d])/4)^n + (e - c*Sqrt[b] + b*Sqrt[d] - Sqrt[g])*((a - Sqrt[b])*(b + Sqrt[d])/4)^n + (e + c*Sqrt[b] - b*Sqrt[d] - Sqrt[g])*((a + Sqrt[b])*(b - Sqrt[d])/4)^n + (e - c*Sqrt[b] - 5*Sqrt[d] + Sqrt[g])*((a - Sqrt[b])*(b - Sqrt[d])/4)^n)/ 140]; Array[f, 17, 0] (* Robert G. Wilson v, Aug 13 2011 *)
a[n_] := (MatrixPower[{{15, -32, 15, -1}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}}, n].{8, 1, 1, 8})[[2]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 31 2016, after Alois P. Heinz *)
a[0] = 1; a[n_] := Product[2(2+Cos[j Pi/3]+Cos[2k Pi/(2n+1)]), {k, 1, n}, {j, 1, 2}] // Round;
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Aug 20 2018 *)
LinearRecurrence[{15, -32, 15, -1}, {1, 8, 95, 1183}, 20] (* Vincenzo Librandi, Aug 20 2018 *)

If b(n) denotes the number of perfect matchings (or domino tilings) in P_5 X P_n we have:
b(1) = 0,
b(2) = 8,
b(3) = 0,
b(4) = 95,
b(5) = 0,
b(6) = 1183,
b(7) = 0,
b(8) = 14824 and
b(n) = 15b(n-2) - 32b(n-4) + 15b(n-6) - b(n-8).
G.f.: (1-x)*(1 - 6*x + x^2)/(1 - 15*x + 32*x^2 - 15*x^3 + x^4).
Let M be the 4 X 4 matrix |1 0 2 8 | 0 1 0 2 | 2 1 5 21| 1 1 1 8 |; then a(n) = M^n(4, 4). - Philippe Deléham, Aug 08 2003
Limit_{n -> infinity} a(n)/a(n-1) = (3 + sqrt(5))*(5 + sqrt(21))/4 = 12.54375443458... - Philippe Deléham, Jun 13 2005
a(n) = ((35 + 7*sqrt(5) + 5*sqrt(21) + sqrt(105))*((3+sqrt(5))*(5+sqrt(21))/4)^n + (35 - 7*sqrt(5) + 5*sqrt(21) - sqrt(105))*((3-sqrt(5))*(5+sqrt(21))/4)^n + (35 + 7*sqrt(5) - 5*sqrt(21) - sqrt(105))*((3+sqrt(5))*(5-sqrt(21))/4)^n + (35 - 7*sqrt(5) - 5*sqrt(21) + sqrt(105))*((3-sqrt(5))*(5-sqrt(21))/4)^n)/140. - Tim Monahan, Aug 13 2011

Recurrence from Faase's web page added by N. J. A. Sloane, Feb 03 2009

A300060 Number of domino tilings of the diagram of the integer partition with Heinz number n.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 0, 2, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 3, 0, 3, 1, 1, 0, 0, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 0, 1, 0, 0, 1, 0, 1, 5, 0, 0, 1, 1, 0, 3, 0, 2, 0, 0, 0, 1, 1, 3, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 4, 1, 0, 0, 1, 0, 5, 1, 0, 2, 3, 0, 2, 1, 1, 1, 5, 0, 0, 1, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Feb 23 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Alois P. Heinz, Table of n, a(n) for n = 1..100000
Wikipedia, Domino tiling
Gus Wiseman, The a(91) = 5 domino tilings of (6,4).

Cf. A000085, A000720, A000712, A000898, A001222, A004003, A056239, A099390, A138178, A153452, A238690, A296150, A296188, A299925, A299926, A300056, A300061, A304662.

h:= proc(l, f) option remember; local k; if min(l[])>0 then
     `if`(nops(f)=0, 1, h(map(x-> x-1, l[1..f[1]]), subsop(1=[][], f)))
    else for k from nops(l) while l[k]>0 by -1 do od;
        `if`(nops(f)>0 and f[1]>=k, h(subsop(k=2, l), f), 0)+
        `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f), 0)
      fi
    end:
g:= l-> `if`(add(`if`(l[i]::odd, (-1)^i, 0), i=1..nops(l))=0,
        `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l))), 0):
a:= n-> g(sort(map(i-> numtheory[pi](i[1])$i[2], ifactors(n)[2]), `>`)):
seq(a(n), n=1..120);  # Alois P. Heinz, May 22 2018

h[l_, f_] := h[l, f] = Module[{k}, If[Min[l] > 0, If[Length[f] == 0, 1, h[Map[Function[x, x-1], l[[Range @ f[[1]]]]], ReplacePart[f, 1 -> Nothing]]], For[k = Length[l], l[[k]] > 0, k-- ]; If[Length[f] > 0 && f[[1]] >= k, h[ReplacePart[l, k -> 2], f], 0] + If[k > 1 && l[[k-1]] == 0, h[ReplacePart[l, {k -> 1, k - 1 -> 1}], f], 0]]];
g[l_] := If[Sum[If[OddQ @ l[[i]], (-1)^i, 0], {i, 1, Length[l]}] == 0, If[l == {}, 1, h[Table[0, l[[1]]], ReplacePart[l, 1 -> Nothing]]], 0];
a[n_] := g[Reverse @ Sort[ Flatten[ Map[ Function[i, Table[PrimePi[i[[1]]], i[[2]]]], FactorInteger[n]]]]];
Array[a, 120] (* Jean-François Alcover, May 28 2018, after Alois P. Heinz *)

A187616 Triangle T(m,n) read by rows: number of domino tilings of the m X n grid (0 <= m <= n).

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 0, 3, 0, 1, 1, 5, 11, 36, 1, 0, 8, 0, 95, 0, 1, 1, 13, 41, 281, 1183, 6728, 1, 0, 21, 0, 781, 0, 31529, 0, 1, 1, 34, 153, 2245, 14824, 167089, 1292697, 12988816, 1, 0, 55, 0, 6336, 0, 817991, 0, 108435745, 0, 1, 1, 89, 571, 18061, 185921, 4213133, 53175517, 1031151241, 14479521761, 258584046368

Offset: 0

N. J. A. Sloane, Mar 11 2011

A099390 is the main entry for this problem.

Triangle read by rows: the square array in A187596 with entries above main diagonal deleted.

			Triangle begins:
1
1 0
1 1  2
1 0  3   0
1 1  5  11   36
1 0  8   0   95     0
1 1 13  41  281  1183   6728
1 0 21   0  781     0  31529       0
1 1 34 153 2245 14824 167089 1292697 12988816
...

Alois P. Heinz, Rows n = 1..32, flattened
Index entries for sequences related to dominoes

Cf. A099390, A187596. See A099390 for sequences appearing in the rows and columns. See also A187617, A187618.

with(LinearAlgebra):
T:= proc(m,n) option remember; local i, j, t, M;
      if m<=1 or n<=1 then 1 -irem(n*m, 2)
    elif irem(n*m, 2)=1 then 0
    else M:= Matrix(n*m, shape =skewsymmetric);
         for i to n do
           for j to m do
             t:= (i-1)*m+j;
             if jAlois P. Heinz, Apr 11 2011

T[m_, n_] := T[m, n] = Module[{i, j, t, M}, Which[m <= 1 || n <= 1, 1 - Mod[n*m, 2], Mod[n*m, 2] == 1, 0, True, M[i_, j_] /; j < i := -M[j, i]; M[, ] = 0; For[i = 1, i <= n, i++, For[j = 1, j <= m, j++, t = (i-1)*m+j; If[j < m, M[t, t+1] = 1]; If[i < n, M[t, t+m] = 1 - 2*Mod[j, 2]]]]; Sqrt[Det[Table[M[i, j], {i, 1, n*m}, {j, 1, n*m}]]]]]; Table[Table[T[m, n], {n, 0, m}], {m, 0, 10}] // Flatten (* Jean-François Alcover, Jan 07 2014, translated from Maple *)

A286912 Array read by antidiagonals: T(m,n) = number of edge covers in the grid graph P_m X P_n.

Original entry on oeis.org

0, 1, 1, 1, 7, 1, 2, 43, 43, 2, 3, 277, 969, 277, 3, 5, 1777, 23663, 23663, 1777, 5, 8, 11407, 571099, 2180738, 571099, 11407, 8, 13, 73219, 13807469, 198906617, 198906617, 13807469, 73219, 13, 21, 469981, 333735575, 18169793971, 68534828391, 18169793971, 333735575, 469981, 21

Offset: 1

Andrew Howroyd, May 15 2017

			Table starts:
======================================================================
m\n| 1     2        3           4              5                 6
---|------------------------------------------------------------------
1  | 0     1        1           2              3                 5 ...
2  | 1     7       43         277           1777             11407 ...
3  | 1    43      969       23663         571099          13807469 ...
4  | 2   277    23663     2180738      198906617       18169793971 ...
5  | 3  1777   571099   198906617    68534828391    23650967140325 ...
6  | 5 11407 13807469 18169793971 23650967140325 30833670159649637 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..276
Eric Weisstein's World of Mathematics, Edge Cover
Eric Weisstein's World of Mathematics, Grid Graph

Rows 1-3 are A000045(n-1), A286911, A288031.
Main diagonal is A286913.
Cf. A210662, A099390, A089934, A218354.

T(1,1) corrected by Andrew Howroyd, Jun 04 2017

A028469 Number of perfect matchings in graph P_{7} X P_{2n}.

Original entry on oeis.org

1, 21, 781, 31529, 1292697, 53175517, 2188978117, 90124167441, 3710708201969, 152783289861989, 6290652543875133, 259009513044645817, 10664383939345916681, 439092316687230373293, 18079062471131097321077

Offset: 0

Per H. Lundow

nonn

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

Alois P. Heinz, Table of n, a(n) for n = 0..500
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
Per Hakan Lundow, Computation of matching polynomials and the number of 1-factors in polygraphs, Research report, No 12, 1996, Department of Math., Umea University, Sweden.
Per Hakan Lundow, Enumeration of matchings in polygraphs, 1998.
R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], Table 6.
Index entries for linear recurrences with constant coefficients, signature (56, -672, 2632, -4094, 2632, -672, 56, -1).

Row 7 of array A099390.

a[n_] := Product[2(2+Cos[k Pi/4]+Cos[2j Pi/(2n+1)]), {k, 1, 3}, {j, 1, n}] // Round;
Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Aug 20 2018 *)

G.f.: (-x^7 +35*x^6 -277*x^5 +727*x^4 -727*x^3 +277*x^2 -35*x +1) / (x^8 -56*x^7 +672*x^6 -2632*x^5 +4094*x^4 -2632*x^3 +672*x^2 -56*x +1).
(Faase:) If b(n) denotes the number of perfect matchings in P_7 X P_n we have:
b(1) = 0,
b(2) = 21,
b(3) = 0,
b(4) = 781,
b(5) = 0,
b(6) = 31529,
b(7) = 0,
b(8) = 1292697,
b(9) = 0,
b(10) = 53175517,
b(11) = 0,
b(12) = 2188978117,
b(13) = 0,
b(14) = 90124167441,
b(15) = 0,
b(16) = 3710708201969, and
b(n) = 56b(n-2) - 672b(n-4) + 2632b(n-6) - 4094b(n-8) + 2632b(n-10) - 672b(n-12) + 56b(n-14) - b(n-16).

Added recurrence from Faase's web page. - N. J. A. Sloane, Feb 03 2009

cp's OEIS Frontend

A065072 Number of ways to tile a square of side 2n by dominoes (rectangles of size 2 X 1 or 1 X 2) is 2^n * a(n)^2 (see A004003).

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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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A187596 Array T(m,n) read by antidiagonals: number of domino tilings of the m X n grid (m>=0, n>=0).

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A239264 Number A(n,k) of domicule tilings of a k X n grid; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A187617 Array T(m,n) read by antidiagonals: number of domino tilings of the 2m X 2n grid (m>=0, n>=0).

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A003775 Number of perfect matchings (or domino tilings) in P_5 X P_2n.

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A300060 Number of domino tilings of the diagram of the integer partition with Heinz number n.

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