A004003 -id:A004003 - cp's OEIS frontend

A054344 Number of ways of covering a 2n X 2n lattice with 2n^2 dominoes of which exactly 6 are horizontal (or vertical) dominoes.

9, 1064, 21656, 197484, 1143366, 4927524, 17240292, 51631617, 137044523, 330284988, 735542444, 1533609350, 3024043008, 5684167992, 10249533240, 17821214019, 30006185613, 49097892704, 78305096016

Offset: 2

Yong Kong (ykong(AT)curagen.com), May 06 2000

			a(3) = 1064 because we have 1064 ways to cover a 36 X 36 lattice with exactly 6 horizontal (or vertical) dominoes and exactly 12 vertical (or horizontal) dominoes.

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
M. E. Fisher, Statistical mechanics of dimers on a plane lattice, Physical Review, 124 (1961), 1664-1672.
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.
Index entries for sequences related to dominoes
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A004003, A002414, A038758.

CoefficientList[Series[(x^9-10*x^8+45*x^7-36*x^6+3096*x^5 +17256*x^4 +27724*x^3+11421*x^2+974*x+9)/(x-1)^10,{x,0,30}],x] (* Vincenzo Librandi, Jun 26 2012 *)

a(n) = (1/720)*n*(n+1)*(120*n^7-300*n^6-70*n^5+363*n^4+416*n^3-231*n^2-394*n-264).

G.f.: x^2*(x^9-10*x^8+45*x^7-36*x^6+3096*x^5+17256*x^4+27724*x^3+11421*x^2+974*x+9)/(x-1)^10. - Colin Barker, Jun 26 2012

A124997 Number of fault-free domino tilings (or dimer coverings) of a 2n X 2n square.

Original entry on oeis.org

0, 0, 0, 25506, 1759280998, 854818404562894, 3588226034666378581610, 138311081613064367684548901556, 50272239752141442901464758051467073726, 174927321882862834702052846250836696969014873138, 5889117928937174007411459040006660524033737246962655301188

Offset: 1

Don Knuth, Jun 26 2008

nonn

Enumerates a subclass of the things counted by A004003.

"Fault-free" means that every straight line through the interior of the square hits the interior of some domino.

Xu Mingkuan, Table of n, a(n) for n = 1..12

Cf. A004003.

a(7) from Alois P. Heinz, Oct 19 2015

a(8)-a(11) from Xu Mingkuan, Dec 23 2016

A127606 a(n) = 2^(2nn) * Product_{1<=i,j<=n} (cos(iPi/(2n+1))^2 + sin(jPi/(2n+1))^2).

Original entry on oeis.org

1, 4, 176, 79808, 372713728, 17931360207872, 8887976555024756736, 45390122553039546330628096, 2388340820825093234015277927170048, 1294826675280341699389150405743029631844352

Offset: 0

Miklos Kristof, Apr 03 2007

nonn

a(n)/4^n is an integer. - Seiichi Manyama, Dec 31 2020

Seiichi Manyama, Table of n, a(n) for n = 0..40
Wikipedia, Chebyshev polynomials
Wikipedia, Resultant

Cf. A004003, A127605.

for n from 0 to 12 do a[n]:=2^(2*n*n)*product(product(cos(i*Pi/(2*n+1))^2+ sin(j*Pi/(2*n+1))^2,j=1..n),i=1..n) od: seq(round(evalf(a[n],300)),n=0..12);

Table[2^(2*n^2) * Product[Product[Cos[i*Pi/(2*n + 1)]^2 + Sin[j*Pi/(2*n + 1)]^2, {i, 1, n}], {j, 1, n}], {n, 0, 15}] // Round (* Vaclav Kotesovec, Mar 18 2023 *)

default(realprecision, 120);
{a(n) = round(prod(i=1, n, prod(j=1, n, 4*cos(i*Pi/(2*n+1))^2+4*sin(j*Pi/(2*n+1))^2)))} \\ Seiichi Manyama, Dec 31 2020

{a(n) = sqrtint(4^n*polresultant(polchebyshev(2*n+1, 1, I*x/2), polchebyshev(2*n, 2, x/2)))} \\ Seiichi Manyama, Jan 09 2021

from math import isqrt
from sympy.abc import x
from sympy import resultant, chebyshevt, chebyshevu, I
def A127606(n): return isqrt(resultant(chebyshevt((n<<1)+1,I*x/2),chebyshevu(n<<1,x/2)))<Chai Wah Wu, Nov 07 2023

a(n) = 2^n * sqrt(Resultant(T_{2*n+1}(i*x/2), U_{2*n}(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, Jan 09 2021

a(n) ~ 2^(1/8) * exp(G*(2*n + 1)^2/Pi) / (1 + sqrt(2))^(n + 1/2), where G is Catalan's constant A006752. - Vaclav Kotesovec, Mar 18 2023

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

John Mason, Nov 17 2017

See links.

			Upper left corner of array:
0,  2,  0,  4,  0, ...
2,  6, 16, 44, ...
0, 16,  0, ...
4, 44, ...
0, ...
...

John Mason, A theorem about unambiguously decomposable rectangular patterns
John Mason, Examples of decomposable patterns

Cf. A295215 (unambiguously tilable patterns), A295216 (ambiguously tilable patterns), A004003 (domino tiling of a square), A099390 (domino tiling of a rectangle).

a(n) = A295215(n) + A295216(n).

A353777 Number of tilings of an n X n square using dominoes, monominoes and 2 X 2 tiles.

Original entry on oeis.org

1, 1, 8, 163, 15623, 5684228, 8459468955, 50280716999785, 1202536689448371122, 115462301811597894998929, 44537596159273736617786474211, 69003082378039459280864860681919942, 429429579883061866326542598342441907826951, 10734684843612889640707750537898705644071715970757

Offset: 0

Alois P. Heinz, May 07 2022

nonn

			a(2) = 8:
  .___.  .___.  .___.  .___.  .___.  .___.  .___.  .___.
  |   |  |_|_|  |___|  | | |  |_|_|  |___|  |_| |  | |_|
  |___|  |_|_|  |___|  |_|_|  |___|  |_|_|  |_|_|  |_|_| .

Alois P. Heinz, Table of n, a(n) for n = 0..17
Wikipedia, Polyomino

Main diagonal of A352589.

Cf. A004003, A028420, A063443, A219874, A219952, A219994, A220061, A220778, A233807, A270071.

a(n) = A352589(n,n).

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

Scott R. Shannon, Feb 18 2023

All possible tilings are counted, including those identical by symmetry. Note that distinct height x width dimensions means that, for example, a 1 x 3 oblong can be used twice, once in a horizonal (1 x 3) and once in a vertical (3 x 1) direction.

			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.

Cf. A360256 (rectangles), A360499, A360498, A182275, A004003, A099390, A065072.

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

Scott R. Shannon and N. J. A. Sloane, Feb 20 2023

All possible tilings are counted, including those identical by symmetry. Note that distinct dimensions means that, for example, a 1 x 3 rectangle can only be used once, regardless of if it lies horizontally or vertically.

Only squares with even edges lengths are possible since the area of a square with odd edge lengths is odd, while the perimeter of any rectangle is even.

			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.

Cf. A360499, A360498, A360725, A360256, A182275, A004003, A099390, A065072.

A060635 a(n) is the number of 2 X 1 domino tilings of the set S in the plane R^2 consisting of the union of the following two rectangles: rectangle1: |x| <= n, |y| <= 1, rectangle2: |x| <= 1, |y| <= n.

Original entry on oeis.org

2, 8, 72, 450, 3200, 21632, 149058, 1019592, 6993800, 47922050, 328499712, 2251473408, 15432082562, 105772401800, 724976569800, 4969058770242, 34058447431808, 233440040239232, 1600021920672450, 10966713178192200, 75166970919070472, 515202081704384258, 3531247605071972352

Offset: 1

Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 16 2001

nonn

The relevant graph has rotational symmetry so the number of tilings is a square or twice a square, in this case by the formula for a(n) it is always twice a square.

			a(1) = 2 because in this case the set S is the unit square and there is one horizontal tiling and one vertical.

Harry J. Smith, Table of n, a(n) for n = 1..200
M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97.
W. Jockusch, Perfect matchings and perfect squares J. Combin. Theory Ser. A 67 (1994), no. 1, 100-115.
Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

Cf. A000045, A001654, A006253, A004003, A006125, A002878, A049658, A197424.

with(combinat): for n from 1 to 40 do printf(`%d,`,2*fibonacci(n)^2*fibonacci(n+1)^2) od:

2*Times @@@ Partition[Fibonacci[Range[25]]^2, 2, 1] (* Paolo Xausa, Jul 03 2025 *)

{ a=1; b=0; c=1; for (n=1, 200, f=a+b; g=b+c; a=b; b=c; c=g; write("b060635.txt", n, " ", 2*f^2*g^2); ) } \\ Harry J. Smith, Jul 08 2009

a(n) = 2 * F(n)^2 * F(n+1)^2 where F(n) is the n-th Fibonacci number - sequence A000045.
G.f.: -2*x*(1-x+x^2) / ( (x-1)*(x^2+3*x+1)*(x^2-7*x+1) ). - R. J. Mathar, Jan 30 2011
a(n) = -4*(-1)^n*A002878(n)/25 - 2/25 + 6*A049658(n)/25. - R. J. Mathar, Jan 30 2011
a(n) = 2 * A001654(n)^2 = 2 * A197424(n-2) for n>=2. - Alois P. Heinz, Jul 03 2025

More terms from James Sellers, Apr 16 2001

A137308 Number of dimer coverings on n X n square if n is even; number of dimer arrangements with exactly one monomer if n is odd.

Original entry on oeis.org

1, 1, 2, 18, 36, 2180, 6728, 2200776, 12988816, 20355006224, 258584046368, 1801272981919008, 53060477521960000, 1560858753560238398528, 112202208776036178000000, 13428038397958481723104394368

Offset: 0

Steven Finch, Apr 03 2008

nonn

Subsequence of even-subscripted terms is A004003.
Rightmost diagonal of A242861.
Also the number of maximum matchings in the n X n grid graph. - Eric W. Weisstein, May 28 2017

Y. Kong, Packing dimers on (2p+1) X (2q+1) lattices, Phys. Rev. E 73 (2006) 016106

Steven Finch, Table of n, a(n) for n = 0..19
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Matching
Eric Weisstein's World of Mathematics, Maximum Independent Edge Set

Cf. A004003, A242861.

a(0)=1 prepended by Alois P. Heinz, Aug 15 2021

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

Alois P. Heinz, Apr 15 2011

nonn

			a(3) = 4 because there are 4 domino tilings of the 3 X 3 grid with upper left corner removed:
  . .___.  . .___.  . .___.  . .___.
  ._|___|  ._|___|  ._| | |  ._|___|
  | |___|  | | | |  | |_|_|  |___| |
  |_|___|  |_|_|_|  |_|___|  |___|_|

Alois P. Heinz, Table of n, a(n) for n = 0..50
Index entries for sequences related to dominoes

Main diagonal of A189006.

Bisection gives: A004003 (even part), A007341 (odd part).

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 *)

a(n) = A189006(n,n).

cp's OEIS Frontend

A054344 Number of ways of covering a 2n X 2n lattice with 2n^2 dominoes of which exactly 6 are horizontal (or vertical) dominoes.

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A124997 Number of fault-free domino tilings (or dimer coverings) of a 2n X 2n square.

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A127606 a(n) = 2^(2*n*n) * Product_{1<=i,j<=n} (cos(i*Pi/(2*n+1))^2 + sin(j*Pi/(2*n+1))^2).

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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.

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Author

Keywords

Comments

Examples

Links

Crossrefs

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A353777 Number of tilings of an n X n square using dominoes, monominoes and 2 X 2 tiles.

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Author

Keywords

Examples

Links

Crossrefs

Formula

A360725 Number of ways to tile an n X n square using oblongs with distinct height x width dimensions.

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Author

Keywords

Comments

Examples

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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.

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Comments

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A060635 a(n) is the number of 2 X 1 domino tilings of the set S in the plane R^2 consisting of the union of the following two rectangles: rectangle1: |x| <= n, |y| <= 1, rectangle2: |x| <= 1, |y| <= n.

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Examples

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Programs

Maple

Mathematica

PARI

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Extensions

A137308 Number of dimer coverings on n X n square if n is even; number of dimer arrangements with exactly one monomer if n is odd.

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A127606 a(n) = 2^(2nn) * Product_{1<=i,j<=n} (cos(iPi/(2n+1))^2 + sin(jPi/(2n+1))^2).