A028468
Number of perfect matchings in graph P_{6} X P_{n}.
Original entry on oeis.org
1, 1, 13, 41, 281, 1183, 6728, 31529, 167089, 817991, 4213133, 21001799, 106912793, 536948224, 2720246633, 13704300553, 69289288909, 349519610713, 1765722581057, 8911652846951, 45005025662792, 227191499132401, 1147185247901449, 5791672851807479
Offset: 0
- 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..450
- 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.
- 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], 2013, Table 5.
- 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 linear recurrences with constant coefficients, signature (1,20,10,-38,-10,20,-1,-1).
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m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (x^2-2*x-1)*(x^4+2*x^3-3*x^2-2*x+1)/((1-x^2)*(x^3-5*x^2+6*x-1)*(x^3+ 6*x^2+5*x+1)) )); // G. C. Greubel, Nov 25 2018
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seq(coeff(series((1+2*x-x^2)*(x^4+2*x^3-3*x^2-2*x+1)/((x-1)*(x+1)*(x^3-5*x^2+6*x-1)*(x^3+6*x^2+5*x+1)),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Nov 23 2018
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a[n_] := Product[2(2 + Cos[(2 k Pi)/7] + Cos[(2 j Pi)/(n+1)]), {k, 1, 3}, {j, 1, n/2}] // Round;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 19 2018, after A099390 *)
LinearRecurrence[{1, 20, 10, -38, -10, 20, -1, -1}, {1, 1, 13, 41, 281, 1183, 6728, 31529}, 30] (* Vincenzo Librandi, Nov 24 2018 *)
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my(x='x+O('x^30)); Vec(-(x^2-2*x-1)*(x^4+2*x^3-3*x^2-2*x+1)/((x-1)*(1+x)*(x^3-5*x^2+6*x-1)*(x^3+6*x^2+5*x+1))) \\ Altug Alkan, Mar 23 2016
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s=((x^2-2*x-1)*(x^4+2*x^3-3*x^2-2*x+1)/((1-x^2)*(x^3-5*x^2+6*x-1) *(x^3+6*x^2+5*x+1))).series(x,30); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 25 2018
A210724
Number of domino tilings of the 11 X n grid with upper left corner removed iff n is odd.
Original entry on oeis.org
1, 1, 144, 780, 51205, 380160, 21001799, 170537640, 8940739824, 74795194705, 3852472573499, 32565539635200, 1666961188795475, 14143261515284447, 722364079570222320, 6136973985625588560, 313196612952258199679, 2662079368040434932480, 135818983640055277506397
Offset: 0
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A[1, 1] = 1;
A[m_, n_] := A[m, n] = Module[{i, j, s, t, M}, Which[m == 0 || n == 0, 1, m < n, A[n, m], True, s = Mod[n*m, 2]; 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 - s; If[i > 1 || j > 1 || s == 0, If[j < m, M[t, t + 1] = 1]; If[i < n, M[t, t + m] = 1 - 2*Mod[j, 2]]]]]; Sqrt[Det[Array[M, {n*m - s, n*m - s}]] ]]];
a[n_] := A[11, n];
a /@ Range[0, 18] (* Jean-François Alcover, Feb 27 2020, after Alois P. Heinz in A189006 *)
A189003
Number of domino tilings of the 5 X n grid with upper left corner removed iff n is odd.
Original entry on oeis.org
1, 1, 8, 15, 95, 192, 1183, 2415, 14824, 30305, 185921, 380160, 2332097, 4768673, 29253160, 59817135, 366944287, 750331584, 4602858719, 9411975375, 57737128904, 118061508289, 724240365697, 1480934568960, 9084693297025, 18576479568193, 113956161827912
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..550
- Index entries for sequences related to dominoes
- Index entries for linear recurrences with constant coefficients, signature (0,15,0,-32,0,15,0,-1).
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a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-1|15|-32|15>>^iquo(n, 2, 'r').
`if`(r=0, <<8, 1, 1, 8>>, <<1, 0, 1, 15>>))[3, 1]:
seq(a(n), n=0..30);
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a[n_] := Product[2(2+Cos[2 j Pi/(n+1)]+Cos[k Pi/3]), {k, 1, 2}, {j, 1, n/2} ] // Round;
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Aug 19 2018, after A099390 *)
A189004
Number of domino tilings of the 7 X n grid with upper left corner removed iff n is odd.
Original entry on oeis.org
1, 1, 21, 56, 781, 2415, 31529, 100352, 1292697, 4140081, 53175517, 170537640, 2188978117, 7022359583, 90124167441, 289143013376, 3710708201969, 11905151192865, 152783289861989, 490179860527896, 6290652543875133
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..400
- Index entries for sequences related to dominoes
- Index entries for linear recurrences with constant coefficients, signature (0, 56, 0, -672, 0, 2632, 0, -4094, 0, 2632, 0, -672, 0, 56, 0, -1).
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A[m_, n_] := A[m, n] = Module[{i, j, s, t, M}, Which[m == 0 || n == 0, 1, m < n, A[n, m], True, s = Mod[n*m, 2]; 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 - s; If[i > 1 || j > 1 || s == 0, If[j < m, M[t, t + 1] = 1]; If[i < n, M[t, t + m] = 1 - 2*Mod[j, 2]]]]]; Sqrt[Det[Array[M, {n*m - s, n*m - s}]] ]]];
a[n_] := A[7, n];
a /@ Range[0, 20] (* Jean-François Alcover, Feb 27 2020, after Alois P. Heinz in A189006 *)
A189005
Number of domino tilings of the 9 X n grid with upper left corner removed iff n is odd.
Original entry on oeis.org
1, 1, 55, 209, 6336, 30305, 817991, 4140081, 108435745, 557568000, 14479521761, 74795194705, 1937528668711, 10021992194369, 259423766712000, 1342421467113969, 34741645659770711, 179796299139278305, 4652799879944138561
Offset: 0
-
A[1, 1] = 1;
A[m_, n_] := A[m, n] = Module[{i, j, s, t, M}, Which[m == 0 || n == 0, 1, m < n, A[n, m], True, s = Mod[n*m, 2]; 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 - s; If[i > 1 || j > 1 || s == 0, If[j < m, M[t, t + 1] = 1]; If[i < n, M[t, t + m] = 1 - 2*Mod[j, 2]]]]]; Sqrt[Det[Array[M, {n*m - s, n*m - s}]] ]]];
a[n_] := A[9, n];
a /@ Range[0, 18] (* Jean-François Alcover, Feb 27 2020, after Alois P. Heinz in A189006 *)
A189002
Number of domino tilings of the n X n grid with upper left corner removed iff n is odd.
Original entry on oeis.org
1, 1, 2, 4, 36, 192, 6728, 100352, 12988816, 557568000, 258584046368, 32565539635200, 53060477521960000, 19872369301840986112, 112202208776036178000000, 126231322912498539682594816, 2444888770250892795802079170816, 8326627661691818545121844900397056
Offset: 0
a(3) = 4 because there are 4 domino tilings of the 3 X 3 grid with upper left corner removed:
. .___. . .___. . .___. . .___.
._|___| ._|___| ._| | | ._|___|
| |___| | | | | | |_|_| |___| |
|_|___| |_|_|_| |_|___| |___|_|
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A[1, 1] = 1;
A[m_, n_] := A[m, n] = Module[{i, j, s, t, M}, Which[m == 0 || n == 0, 1, m < n, A[n, m], True, s = Mod[n*m, 2]; 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 - s; If[i > 1 || j > 1 || s == 0, If[j < m, M[t, t + 1] = 1]; If[i < n, M[t, t + m] = 1 - 2*Mod[j, 2]]]]]; Sqrt[Det[Array[M, {n*m - s, n*m - s}]] ]]];
a[n_] := A[n, n];
a /@ Range[0, 17] (* Jean-François Alcover, Feb 27 2020, after Alois P. Heinz in A189006 *)
A270668
Triangle read by rows: The number of domino tilings of the (2n+1) X (2m+1) board with a central free square.
Original entry on oeis.org
1, 0, 2, 1, 0, 196, 0, 32, 0, 75272, 1, 0, 31329, 0, 599466256, 0, 450, 0, 135663392, 0, 28838245503008, 1, 0, 4941729, 0, 10956424382401, 0, 22463213552677201984, 0, 6272, 0, 233075146752, 0, 5652453608244879872, 0, 123818965842734619629420672
Offset: 0
For n=m=1, the 3 X 3 board can be covered in T(1,1)=2 ways, starting in one corner with either a horizontal or a vertical domino.
Triangle begins:
1;
0, 2;
1, 0, 196;
0, 32, 0, 75272;
1, 0, 31329, 0, 599466256;
0, 450, 0, 135663392, 0, 28838245503008;
1, 0, 4941729, 0, 10956424382401, 0, 22463213552677201984;
Showing 1-7 of 7 results.
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