A295214
Array T(m,n) read by antidiagonals: number of m X n rectangular patterns of precisely half black squares and half white squares that are tilable with black and white colored dominoes, for m >= 1, n >= 1.
Original entry on oeis.org
0, 2, 2, 0, 6, 0, 4, 16, 16, 4, 0, 44, 0, 44, 0, 8, 120, 318, 318, 120, 8, 0, 328, 0, 2798, 0, 328, 0, 16, 896, 6334, 22222, 22222, 6334, 896, 16
Offset: 1
Upper left corner of array:
0, 2, 0, 4, 0, ...
2, 6, 16, 44, ...
0, 16, 0, ...
4, 44, ...
0, ...
...
Cf.
A295215 (unambiguously tilable patterns),
A295216 (ambiguously tilable patterns),
A004003 (domino tiling of a square),
A099390 (domino tiling of a rectangle).
A295215
Array T(m,n) read by antidiagonals: number of m X n rectangular patterns of precisely half black squares and half white squares that are unambiguously tilable with black and white colored dominoes, for m >= 1, n >= 1.
Original entry on oeis.org
0, 2, 2, 0, 4, 0, 4, 10, 10, 4, 0, 22, 0, 22, 0, 8, 50, 144, 144, 50, 8, 0, 114, 0, 864, 0, 114, 0, 16, 258, 1924, 5354, 5354, 1924, 258, 16
Offset: 1
Upper left corner of array:
0, 2, 0, 4, 0, ...
2, 4, 10, 22, ...
0, 10, 0, ...
4, 22, ...
0, ...
...
Cf.
A295214 for all tilable patterns,
A295216 for ambiguously tilable patterns,
A099390 for domino tiling of a rectangle.
A295216
Array T(m,n) read by antidiagonals: number of m X n rectangular patterns of precisely half black squares and half white squares that are ambiguously tilable with black and white colored dominoes, for m >= 1, n >= 1.
Original entry on oeis.org
0, 0, 0, 0, 2, 0, 0, 6, 6, 0, 0, 22, 0, 22, 0, 0, 70, 174, 174, 70, 0, 0, 214, 0, 1934, 0, 214, 0, 0, 638, 4410, 16868, 16868, 4410, 638, 0
Offset: 1
Upper left corner of array:
0, 0, 0, 0, 0, ...
0, 2, 6, 22, ...
0, 6, 0, ...
0, 22, ...
0, ...
...
Cf.
A295214 for all tilable patterns,
A295215 for unambiguously tilable patterns,
A099390 for domino tiling of a rectangle.
A340532
Number of domino tilings of a 16 X n rectangle.
Original entry on oeis.org
1, 1, 1597, 29681, 9475901, 366944287, 69289288909, 3710708201969, 540061286536921, 34741645659770711, 4337452956682508609, 313196612952258199679, 35457442115448212075033, 2764079753958605286860951, 293251198441417290172509377, 24080184063411167042923575793
Offset: 0
a(1) = 1, since there is only one domino tiling of the 16 X n rectangle, which consists entirely of horizontal tiles.
a(2) = 1597 = F(17), since the number of domino tilings of the m X 2 rectangle is the Fibonacci number F(m+1).
Note that the terms a(16) and a(33) are even. More generally, for m even, the numbers of domino tilings of the m X m square and of the m X (2m+1) rectangle are even.
- A. M. Magomedov, T. A. Magomedov, S. A. Lawrencenko, Mutually-recursive formulas for enumerating partitions of the rectangle (Russian, English summary), Prikl. Diskr. Mat., 46 (2019), 108-121.
- Alois P. Heinz, Table of n, a(n) for n = 0..514
- M. E. Fisher, Statistical mechanics of dimers on a plane lattice, Phys. Rev. 124 (1961) 1664-1672.
- P. W. Kasteleyn, The statistics of dimers on a lattice, Physica 27 (1961), 1209-1225.
- P. W. Kasteleyn, Dimer statistics and phase transitions, J. Math. Phys., 4 (1963), 287-293.
- Viet-Ha Nguyen, Kévin Perrot, Mathieu Vallet, NP-completeness of the game Kingdomino(TM), Theoretical Computer Science (2020) Vol. 822, 23-35. See also arXiv:1909.02849, [cs.CC], 2019.
-
b:= proc(n, l) option remember; local k;
if n=0 then 1
elif min(l)>0 then (t-> b(n-t, map(h->h-t, l)))(min(l))
else for k while l[k]>0 do od; `if`(n>1, b(n, subsop(k=2, l)), 0)+
`if`(k b(n, [0$16]):
seq(a(n), n=0..15); # Alois P. Heinz, Jan 12 2021
-
Do[
P = 1; m = 16;
Do[
P = N[P*(4*Cos[Pi*i/(n + 1)]^2 + 4*Cos[Pi*j/(m + 1)]^2), 1020],
{i, 1, n/2}, {j, 1, m/2}];
Print["P=", N[P, 1020], " n=", n, " m=", m],
{n, 1, 20}
]
A360725
Number of ways to tile an n X n square using oblongs with distinct height x width dimensions.
Original entry on oeis.org
0, 0, 4, 36, 1056, 31052, 1473944, 87469884
Offset: 1
a(1) = 0 as no distinct oblongs can tile a square with dimensions 1 x 1.
a(2) = 0 as no distinct oblongs can tile a square with dimensions 2 x 2.
a(3) = 4. There is one tiling, excluding those equivalent by symmetry:
.
+---+---+---+
| |
+---+---+---+
| |
+ +
| |
+---+---+---+
.
This tiling can occur in 4 different ways, giving 4 ways in total.
a(4) = 36. The possible tilings, excluding those equivalent by symmetry, are:
.
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
| | | | | | | | | | |
+ + + +---+---+---+---+ + +---+---+---+ + +---+---+---+
| | | | | | | | | | | |
+---+---+---+---+ + + + + + + + + +
| | | | | | | | | | |
+ + + + +---+---+---+---+ +---+---+ +
| | | | | | | | |
+---+---+---+---+ +---+---+---+---+ +---+---+---+---+ +---+---+---+---+
.
The first tiling can occur in 8 different ways, the second in 4 different ways, the third in 16 different ways and the fourth in 8 different ways. This gives 36 ways in total.
A360773
Number of ways to tile a 2n X 2n square using rectangles with distinct dimensions such that the sum of the rectangles perimeters equals the area of the square.
Original entry on oeis.org
0, 1, 8, 1024, 620448
Offset: 1
a(1) = 0 as a 2 x 2 square, with area 4, cannot be tiled with distinct rectangles with perimeters that sum to 4.
a(2) = 1 as a 4 x 4 rectangle, with area 16, can be tiled with a 4 x 4 square with perimeter 4 + 4 + 4 + 4 = 16.
a(3) = 8. The possible tilings for the 6 x 6 square, with area 36, excluding those equivalent by symmetry, are:
.
+---+---+---+---+---+---+ +---+---+---+---+---+---+
| | | |
+---+---+---+---+---+---+ + +
| | | |
+ + +---+---+---+---+---+---+
| | | |
+ + + +
| | | |
+ + + +
| | | |
+ + + +
| | | |
+---+---+---+---+---+---+ +---+---+---+---+---+---+
.
where for the first tiling (2*6 + 2*1) + (2*6 + 2*5) = 36 while for the second tiling (2*6 + 2*2) + (2*6 + 2*4) = 36. Both of these tilings can occur in 4 ways, giving 8 ways in total.
a(4) = 1024. And example tiling of the 8 x 8 square, with area 64, is:
.
+---+---+---+---+---+---+---+---+
| | | |
+ + +---+---+
| | | |
+ + + +
| | | |
+---+---+---+---+---+---+---+---+
| |
+ +
| |
+ +
| |
+ +
| |
+ +
| |
+---+---+---+---+---+---+---+---+
.
where (2*1 + 2*3) + (2*5 + 2*3) + (2*2 + 2*1) + (2*2 + 2*2) + (2*8 + 2*5) = 64.
A241908
Number of perfect matchings in graph P_{13} X P_{2n}.
Original entry on oeis.org
1, 377, 413351, 536948224, 731164253833, 1012747193318519, 1412218550274852671, 1974622635952709613247, 2764079753958605286860951, 3870940598132705729413670953, 5422065916132126528319352874496, 7595338059193606161156363370300487, 10640045682768766172108553992086690201
Offset: 0
- A. M. Karavaev and S. N. Perepechko, Generating functions for dimer problem on rectangular lattices (in Russian), Information Processes, 13(2013), No4, 374-400.
-
{a(n) = sqrtint(polresultant(polchebyshev(2*n, 2, x/2), polchebyshev(13, 2, I*x/2)))} \\ Seiichi Manyama, Apr 13 2020
A189245
T(n,k) = Number of n X k array permutations with each element moved by a city block distance of one.
Original entry on oeis.org
0, 1, 1, 0, 4, 0, 1, 9, 9, 1, 0, 25, 0, 25, 0, 1, 64, 121, 121, 64, 1, 0, 169, 0, 1296, 0, 169, 0, 1, 441, 1681, 9025, 9025, 1681, 441, 1, 0, 1156, 0, 78961, 0, 78961, 0, 1156, 0, 1, 3025, 23409, 609961, 1399489, 1399489, 609961, 23409, 3025, 1
Offset: 1
R. H. Hardin, with A099390 interpretative help from William Keith in the Sequence Fans Mailing List, Apr 19 2011
Table starts
0, 1, 0, 1, 0, 1, 0
1, 4, 9, 25, 64, 169, 441
0, 9, 0, 121, 0, 1681, 0
1, 25, 121, 1296, 9025, 78961, 609961
0, 64, 0, 9025, 0, 1399489, 0
1, 169, 1681, 78961, 1399489, 45265984, 994077841
0, 441, 0, 609961, 0, 994077841, 0
1,1156, 23409, 5040025, 219750976, 27918733921, 1671065533809
0,3025, 0, 40144896, 0, 669109276081, 0
1,7921,326041,326199721,34566618241.17750489675689,2827635608217289
Some.solutions for 4 X 4:
..4..5..3..2....4..0..3..2....1..2..6..7....4..0..3..7....1..2..3..7
..0..1..7..6....5..1..7..6....0..4..5..3....5..1..2..6....0..9.10..6
..9.10.14.15...12.13.11.10...12.10.11.15....9..8.11.15....4..5.14.15
..8.12.13.11....8..9.15.14....8..9.13.14...13.12.10.14....8.12.13.11
-
T(n, k) = abs(polresultant(polchebyshev(n, 2, x/2), polchebyshev(k, 2, I*x/2))); \\ Seiichi Manyama, Oct 28 2023
A260032
Number of perfect matchings in graph P_{2n} X P_{2n} with a monomer on each corner.
Original entry on oeis.org
1, 8, 784, 913952, 12119367744, 1773206059548800, 2808001509386950713600, 47534638766423741578738188800, 8530835766072904609739799813424153600, 16137081911409285302469685272022812457875802112, 320397648203287990193211938297925486964232264783587250176
Offset: 1
-
with(LinearAlgebra):
a:= proc(n) option remember; local d, i, j, t, m, M;
d:= 2*n; m:= d^2-4;
M:= Matrix(m, shape=skewsymmetric);
for i to d-3 do M[i+1, i]:=1 od;
for i to d-2 do M[i, i+d-1]:=1 od;
for i from m-d+3 to m-1 do M[i, i+1]:=1 od;
for i from m-d+3 to m do M[i-d+1, i]:=1 od;
for i from d-1 to m-2*d+2 do M[i, i+d]:=1 od;
for i to d-2 do for j to d-1 do
t:=d*i+j-2; M[t, t+1]:= `if`(irem(i, 2)=1, 1, -1);
od od;
isqrt(Determinant(M))
end:
seq(a(n), n=1..11); # Alois P. Heinz, Mar 10 2016
-
a[1] = 1; a[n_] := a[n] = Module[{d, i, j, t, m, M}, d = 2*n; m = d^2 - 4; M = Array[0&, {m, m}];
For[i = 1, i <= d - 3, i++, M[[i + 1, i]] = 1];
For[i = 1, i <= d - 2, i++, M[[i, i + d - 1]] = 1];
For[i = m - d + 3, i <= m - 1, i++, M[[i, i + 1]] = 1];
For[i = m - d + 3, i <= m, i++, M[[i - d + 1, i]] = 1];
For[i = d - 1, i <= m - 2*d + 2, i++, M[[i, i + d]] = 1];
For[i = 1, i <= d - 2, i++,
For[j = 1, j <= d - 1, j++, t = d*i + j - 2; M[[t, t + 1]] = If[Mod[i, 2] == 1, 1, -1]]]; M = M - Transpose[M]; Sqrt[Det[M]]];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 11}] (* Jean-François Alcover, Nov 11 2017, after Alois P. Heinz *)
A340535
Number of domino tilings (or dimer coverings) of the 2n X n grid.
Original entry on oeis.org
1, 1, 5, 41, 2245, 185921, 106912793, 90124167441, 540061286536921, 4652799879944138561, 289415868852204573601981, 25545661075321867247577262777, 16457725663617130715785831809325501, 14905470663149838513993965664256435411841, 99323759360556656337166635121447749135517599089
Offset: 0
a(2) = 5:
.___. .___. .___. .___. .___.
|___| |___| |___| | | | | | |
|___| |___| | | | |_|_| |_|_|
|___| | | | |_|_| |___| | | |
|___| |_|_| |___| |___| |_|_|
.
-
b:= proc(m, n) option remember; local i, j, t, M;
M:= Matrix(n*m, shape=skewsymmetric);
for i to n do for j to m do t:= (i-1)*m+j;
if j b(2*n, n):
seq(a(n), n=0..15);
-
T[?OddQ, ?OddQ] = 0;
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}];
a[n_] := T[2n, n] // Round;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 27 2022 *)
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