A065072 -id:A065072 - cp's OEIS frontend

A004003 Number of domino tilings (or dimer coverings) of a 2n X 2n square.

1, 2, 36, 6728, 12988816, 258584046368, 53060477521960000, 112202208776036178000000, 2444888770250892795802079170816, 548943583215388338077567813208427340288, 1269984011256235834242602753102293934298576249856

Offset: 0

N. J. A. Sloane, Greg Huber

A099390 is the main entry for domino tilings (or dimer tilings) of a rectangle.
The numbers of domino tilings in A006253, A004003, A006125 give the number of perfect matchings in the relevant graphs. There are results of Jockusch and Ciucu that if a planar graph has a rotational symmetry then the number of perfect matchings is a square or twice a square - this applies to these 3 sequences. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 12 2001
Christine Bessenrodt points out that Pachter (1997) shows that a(n) is divisible by 2^n (cf. A065072).
a(n) is the number of different ways to cover a 2n X 2n lattice with 2n^2 dominoes. John and Sachs show that a(n) = 2^n*B(n)^2, where B(n) == n+1 (mod 32) when n is even and B(n) == (-1)^((n-1)/2)*n (mod 32) when n is odd. - Yong Kong (ykong(AT)curagen.com), May 07 2000

			The 36 solutions for the 4 X 4 board, from Neven Juric, May 14 2008:
A01 = {(1,2), (3,4), (5,6), (7,8), (9,10), (11,12), (13,14), (15,16)}
A02 = {(1,2), (3,4), (5,6), (7,11), (9,10), (8,12), (13,14), (15,16)}
A03 = {(1,2), (3,4), (5,9), (6,7), (10,11), (8,12), (13,14), (15,16)}
A04 = {(1,2), (3,4), (5,9), (6,10), (7,8), (11,12), (13,14), (15,16)}
A05 = {(1,2), (3,4), (5,9), (6,10), (7,11), (8,12), (13,14), (15,16)}
A06 = {(1,2), (3,4), (5,6), (7,8), (9,10), (13,14), (11,15), (12,16)}
A07 = {(1,2), (3,4), (5,9), (6,10), (7,8), (11,15), (13,14), (12,16)}
A08 = {(1,2), (3,4), (5,6), (7,8), (9,13), (10,14), (11,12), (15,16)}
A09 = {(1,2), (3,4), (5,6), (7,11), (8,12), (9,13), (10,14), (15,16)}
A10 = {(1,2), (3,4), (5,6), (7,8), (9,13), (10,11), (14,15), (12,16)}
A11 = {(1,2), (3,4), (5,6), (7,8), (9,13), (10,14), (11,15), (12,16)}
A12 = {(1,2), (5,6), (3,7), (4,8), (9,10), (11,12), (13,14), (15,16)}
A13 = {(1,2), (3,7), (4,8), (5,9), (6,10), (11,12), (13,14), (15,16)}
A14 = {(1,2), (5,6), (3,7), (4,8), (9,10), (13,14), (11,15), (12,16)}
A15 = {(1,2), (3,7), (4,8), (6,10), (5,9), (11,15), (12,16), (13,14)}
A16 = {(1,2), (3,7), (4,8), (5,6), (9,13), (10,14), (11,12), (15,16)}
A17 = {(1,2), (3,7), (4,8), (5,6), (9,13), (10,11), (14,15), (12,16)}
A18 = {(1,2), (5,6), (3,7), (4,8), (9,13), (10,14), (11,15), (12,16)}
A19 = {(1,5), (2,6), (3,4), (7,8), (9,10), (11,12), (13,14), (15,16)}
A20 = {(1,5), (2,6), (3,4), (7,11), (8,12), (9,10), (13,14), (15,16)}
A21 = {(1,5), (3,4), (2,6), (9,10), (7,8), (11,15), (13,14), (12,16)}
A22 = {(1,5), (2,6), (3,4), (7,8), (9,13), (10,14), (11,12), (15,16)}
A23 = {(1,5), (2,6), (3,4), (7,11), (8,12), (9,13), (10,14), (15,16)}
A24 = {(1,5), (2,6), (3,4), (7,8), (9,13), (10,11), (14,15), (12,16)}
A25 = {(1,5), (2,6), (3,4), (7,8), (9,13), (10,14), (11,15), (12,16)}
A26 = {(1,5), (2,3), (6,7), (4,8), (9,10), (11,12), (13,14), (15,16)}
A27 = {(1,5), (2,6), (3,7), (4,8), (9,10), (11,12), (13,14), (15,16)}
A28 = {(1,5), (2,3), (6,7), (4,8), (9,10), (11,15), (13,14), (12,16)}
A29 = {(1,5), (2,6), (3,7), (4,8), (9,10), (13,14), (11,15), (12,16)}
A30 = {(1,5), (2,3), (6,7), (4,8), (9,13), (10,14), (11,12), (15,16)}
A31 = {(1,5), (2,6), (3,7), (4,8), (9,13), (10,14), (11,12), (15,16)}
A32 = {(1,5), (2,6), (3,7), (4,8), (9,13), (10,14), (11,15), (12,16)}
A33 = {(1,5), (2,6), (3,7), (4,8), (9,13), (10,11), (14,15), (12,16)}
A34 = {(1,5), (2,3), (4,8), (6,10), (7,11), (9,13), (14,15), (12,16)}
A35 = {(1,5), (2,3), (6,7), (4,8), (9,13), (10,14), (11,15), (12,16)}
A36 = {(1,5), (2,3), (6,7), (4,8), (9,13), (10,11), (14,15), (12,16)}

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 569.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 406-412.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Darko Veljan, Kombinatorika s teorijom grafova (Croatian) (Combinatorics with Graph Theory) mentions the value 12988816 = 2^4*901^2 for the 8 X 8 case on page 4.

Alois P. Heinz, Table of n, a(n) for n = 0..44 (first 26 terms from T. D. Noe)
M. Aanjaneya and S. P. Pal, Faultfree tromino tilings of rectangles, arXiv:math/0610925 [math.CO], 2006.
N. Allegra, Exact solution of the 2d dimer model: Corner free energy, correlation functions and combinatorics, arXiv:1410.4131 [cond-mat.stat-mech], 2014.
M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, Journal of Combinatorial Theory, Series A, Volume 77, Issue 1, January 1997, Pages 67-97.
H. Cohn, 2-adic behavior of numbers of domino tilings, arXiv:math/0008222 [math.CO], 2000.
Steven R. Finch, The Dimer Problem [From Steven Finch, Apr 20 2019]
M. E. Fisher, Statistical mechanics of dimers on a plane lattice, Physical Review, 124 (1961), 1664-1672.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 363
Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter, and Tianyuan Xu, Sandpiles and Dominos, Electronic Journal of Combinatorics, Volume 22(1), 2015.
Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter, and Tianyuan Xu, Illustration for a(2) = 36 [Fig. 9 from "Sandpiles and Dominos"]
W. Jockusch, Perfect matchings and perfect squares J. Combin. Theory Ser. A 67 (1994), no. 1, 100-115.
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.
Adrien Kassel, Le modèle de dimères, Images des Mathématiques, CNRS, 2016. [In French]
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.
P. W. Kasteleyn, Dimer statistics and phase transitions, J. Mathematical Phys. 4 1963 287-293. MR0153427 (27 #3394).
Viet-Ha Nguyen, Kévin Perrot, and 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.
Lior Pachter, Combinatorial approaches and conjectures ..., Elec. J. Comb. 4 (1997) #R29.
James Propp, Some 2-Adic Conjectures Concerning Polyomino Tilings of Aztec Diamonds, Integers (2023) Vol. 23, Art. A30.
Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9:3 (2005), 609-640.
N. J. A. Sloane, Illustration for a(2) = 36 [Slide from an old talk I gave]
R. P. Stanley, A combinatorial miscellany
Eric Weisstein's World of Mathematics, Domino Tiling
Eric Weisstein, Illustration for a(2) = 36, from Domino Tilings web page (see previous link) [Included with permission]
Index entries for sequences related to dominoes

Cf. A028420, A006253, A006125, A065072, A124997, A239273, A256043.

Main diagonal of array A099390 or A187596.

f := n->2^(2*n^2)*product(product(cos(i*Pi/(2*n+1))^2+cos(j*Pi/(2*n+1))^2,j=1..n),i=1..n); for k from 0 to 12 do round(evalf(f(k),300)) od;

a[n_] := Round[ N[ Product[ 2*Cos[(2i*Pi)/(2n+1)] + 2*Cos[(2j*Pi)/(2n+1)] + 4,  {i, 1, n}, {j, 1, n}], 300] ]; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Jan 04 2012, after Maple *)
Table[Sqrt[Resultant[ChebyshevU[2*n, x/2], ChebyshevU[2*n, I*x/2], x]], {n, 0, 12}] (* Vaclav Kotesovec, Dec 30 2020 *)

{a(n) = sqrtint(polresultant(polchebyshev(2*n, 2, x/2), polchebyshev(2*n, 2, I*x/2)))} \\ Seiichi Manyama, Apr 13 2020

from math import isqrt
from sympy.abc import x
from sympy import I, resultant, chebyshevu
def A004003(n): return isqrt(resultant(chebyshevu(n<<1,x/2),chebyshevu(n<<1,I*x/2))) if n else 1 # Chai Wah Wu, Nov 07 2023

a(n) = A099390(2n,2n).
a(n) = Product_{j=1..n} Product_{k=1..n} (4*cos(j*Pi/(2*n+1))^2 + 4*cos(k*Pi/(2*n+1))^2). - N. J. A. Sloane, Mar 16 2015
a(n) = 2^n * A065072(n)^2. - Alois P. Heinz, Nov 22 2018
a(n)^2 = Resultant(U(2*n,x/2), U(2*n,i*x/2)), where U(n,x) is a Chebyshev polynomial of the second kind and i = sqrt(-1). - Seiichi Manyama, Apr 13 2020
a(n) ~ 2 * (sqrt(2)-1)^(2*n+1) * exp(G*(2*n+1)^2/Pi), where G is Catalan's constant A006752. - Vaclav Kotesovec, Dec 30 2020

Corrected and extended by David Radcliffe

A007726 Number of spanning trees of quarter Aztec diamonds of order n.

Original entry on oeis.org

1, 1, 4, 56, 2640, 411840, 210613312, 351102230528, 1901049105201408, 33349238079515381760, 1892086487183556298556416, 346728396311328694807284940800, 205021218459835103075295973360128000, 390870571052378289975757743555515137130496

Offset: 1

Richard Stanley

nonn

Mihai Ciucu (ciucu(AT)math.gatech.edu), in preparation, 2001.

Seiichi Manyama, Table of n, a(n) for n = 1..50
Timothy Y. Chow, The Q-spectrum and spanning trees of tensor products of bipartite graphs, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3155-3161.
R. Kenyon, J. Propp and D. Wilson, Trees and matchings, Electronic Journal of Combinatorics, 7(1):R25, 2000.
D. E. Knuth, Aztec Diamonds, Checkerboard Graphs, and Spanning Trees, arXiv:math/9501234 [math.CO], 1995; J. Alg. Combinatorics 6 (1997), 253-257.
R. P. Stanley, Spanning trees of Aztec diamonds, Discrete Math. 157 (1996), 375-388 (Problem 251).
Index entries for sequences related to trees

Cf. A007725, A007341, A065072, A340052.

Table[Product[Product[4 - 2*Cos[j*Pi/n] - 2*Cos[k*Pi/n], {j, 1, k-1}], {k, 2, n-1}], {n, 1, 15}] // Round (* Vaclav Kotesovec, Dec 30 2020 *)
Table[Sqrt[Resultant[ChebyshevU[n-1, x/2], ChebyshevU[n-1, (4-x)/2], x] / (n * 2^(n-1))], {n, 1, 15}] (* Vaclav Kotesovec, Dec 30 2020 *)

default(realprecision, 120);
{a(n) = round(prod(j=2, n-1, prod(i=1, j-1, 4*sin(i*Pi/(2*n))^2+4*sin(j*Pi/(2*n))^2)))} \\ Seiichi Manyama, Dec 29 2020

a(n) = Product_{0Sean A. Irvine, Jan 20 2018
From Vaclav Kotesovec, Dec 30 2020: (Start)
a(n) ~ sqrt(Gamma(1/4)) * 2^(5/8) * exp(2*G*n^2/Pi) / (Pi^(3/8) * n^(3/4) * 2^(n/2) * (1 + sqrt(2))^n), where G is Catalan's constant A006752.
a(n) = sqrt(A007341(n) / (n * 2^(n-1))). (End)

More terms from Sean A. Irvine, Jan 20 2018

A256043 Order of all-2s configuration on the 2n X 2n sandpile grid graph.

Original entry on oeis.org

1, 3, 29, 901, 89893, 5758715, 22687425

Offset: 1

N. J. A. Sloane, Mar 14 2015

Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter, Tianyuan Xu, Sandpiles and Dominos, Illustration of recurrent identity element for the sandpile grid graph [Figure 1 from Sandpiles and Dominos paper] [Le Borgne et al., 2002, show a very similar figure - see Fig. 7.]
Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter, and Tianyuan Xu, Sandpiles and Dominos, Electronic Journal of Combinatorics, Volume 22(1), 2015. Also arXiv:1406.0100, May 31 2014. [Mentions this sequence together with a different sequence (A065072) with the same initial terms]
Le Borgne, Yvan, and Dominique Rossin, On the identity of the sandpile group, Discrete mathematics 256.3 (2002): 775-790.
David Perkinson, Lecture 15: Sandpiles, PCMI 2008 Undergraduate Summer School. Gives a(7).

Cf. A065072, A256047.

Bisection of main diagonal of A256045.

a(7) added by Scott R. Shannon, Apr 23 2019

A360499 Number of ways to tile an n X n square using rectangles with distinct dimensions.

Original entry on oeis.org

1, 1, 21, 269, 4489, 82981, 2995185, 118897973

Offset: 1

Scott R. Shannon, Feb 09 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.

			a(1) = 1 as the only way to tile a 1 x 1 square is with a square with dimensions 1 x 1.
a(2) = 1 as the only way to tile a 2 x 2 square is with a square with dimensions 2 x 2.
a(3) = 21. The possible tilings, excluding those equivalent by symmetry, are:
.
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
  |           |   |   |       |   |       |   |   |           |
  +           +   +---+---+---+   +---+---+   +   +---+---+---+
  |           |   |           |   |       |   |   |           |
  +           +   +           +   +       +   +   +           +
  |           |   |           |   |       |   |   |           |
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
.
The first tiling can occur in 1 way, the second in 8 different ways, the third in 8 different ways and the fourth in 4 different ways, giving 21 ways in total.

Cf. A360498 (oblongs), A182275 (not necessarily distinct dimensions), A004003, A099390, A065072, A233320, A230031.

A340052 a(n) = Product_{1<=i

Original entry on oeis.org

1, 1, 5, 91, 5661, 1173821, 801125065, 1786768287095, 12964564030176889, 305121026002697122873, 23243604301636717923421133, 5722586073277932639539150258131, 4548248834078776410469611991220703125

Offset: 0

Seiichi Manyama, Dec 29 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..50

Cf. A007726, A065072, A127605.

Table[2^(n*(n-1)) * Product[Product[Sin[i*Pi/(2*n + 1)]^2 + Sin[j*Pi/(2*n + 1)]^2, {i, 1, j-1}], {j, 2, n}], {n, 0, 15}] // Round (* Vaclav Kotesovec, Dec 30 2020 *)

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

a(n)^2 = A127605(n)/(2^n * (2*n+1)).

a(n) ~ sqrt(Gamma(1/4)) * exp(G*(2*n+1)^2/(2*Pi)) / (2^(n/2 + 5/4) * Pi^(3/8) * n^(3/4)), where G is Catalan's constant A006752. - Vaclav Kotesovec, Dec 30 2020

A340185 Number of spanning trees in the halved Aztec diamond HOD_n.

Original entry on oeis.org

1, 1, 15, 2639, 5100561, 105518291153, 23067254643457375, 52901008815129395889375, 1266973371422697144030728637409, 315937379766837559600972497421046382689, 818563964325891485548944567913851815851212484079

Offset: 0

Seiichi Manyama, Dec 31 2020

nonn

                                              *
                                              |
                      *                   *---*---*
                      |                   |   |   |
      *           *---*---*           *---*---*---*---*
      |           |   |   |           |   |   |   |   |
  *---*---*   *---*---*---*---*   *---*---*---*---*---*---*
    HOD_1           HOD_2                   HOD_3
-------------------------------------------------------------
                  *
                  |
              *---*---*
              |   |   |
          *---*---*---*---*
          |   |   |   |   |
      *---*---*---*---*---*---*
      |   |   |   |   |   |   |
  *---*---*---*---*---*---*---*---*
                HOD_4

Seiichi Manyama, Table of n, a(n) for n = 0..40
Mihai Ciucu, Symmetry classes of spanning trees of Aztec diamonds and perfect matchings of odd squares with a unit hole, arXiv:0710.4500 [math.CO], 2007. See Corollary 3.7.

Cf. A004003, A007725, A007726, A065072, A127605, A340052, A340176 (halved Aztec diamond HMD_n).

Table[4^((n-1)*n) * Product[Product[(1 - Cos[j*Pi/(2*n + 1)]^2*Cos[k*Pi/(2*n + 1)]^2), {k, j+1, n}], {j, 1, n}], {n, 0, 12}] // Round (* Vaclav Kotesovec, Jan 03 2021 *)

default(realprecision, 120);
{a(n) = round(prod(j=1, 2*n, prod(k=j+1, 2*n-j, 4-4*cos(j*Pi/(2*n+1))*cos(k*Pi/(2*n+1)))))}

default(realprecision, 120);
{a(n) = round(4^((n-1)*n)*prod(j=1, n, prod(k=j+1, n, 1-(cos(j*Pi/(2*n+1))*cos(k*Pi/(2*n+1)))^2)))} \\ Seiichi Manyama, Jan 02 2021

# Using graphillion
from graphillion import GraphSet
def make_HOD(n):
    s = 1
    grids = []
    for i in range(2 * n + 1, 1, -2):
        for j in range(i - 2):
            a, b, c = s + j, s + j + 1, s + i + j
            grids.extend([(a, b), (b, c)])
        grids.append((s + i - 2, s + i - 1))
        s += i
    return grids
def A340185(n):
    if n == 0: return 1
    universe = make_HOD(n)
    GraphSet.set_universe(universe)
    spanning_trees = GraphSet.trees(is_spanning=True)
    return spanning_trees.len()
print([A340185(n) for n in range(7)])

a(n) = Product_{1<=j

From Seiichi Manyama, Jan 02 2021: (Start)

a(n) = 4^((n-1)*n) * Product_{1<=j

a(n) = A340052(n) * A065072(n) = (1/2^n) * sqrt(A127605(n) * A004003(n) / (2*n+1)). (End)

a(n) ~ sqrt(Gamma(1/4)) * exp(G*(2*n+1)^2/Pi) / (Pi^(3/8) * n^(3/4) * 2^(n + 3/4) * (1 + sqrt(2))^(n + 1/2)), where G is Catalan's constant A006752. - Vaclav Kotesovec, Jan 03 2021

A360256 Number of ways to tile an n X n square using rectangles with distinct height X width dimensions.

Original entry on oeis.org

1, 1, 33, 513, 14409, 693025, 50447161

Offset: 1

Scott R. Shannon, Feb 17 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 rectangle can be used twice, once in a horizontal (1 X 3) and once in a vertical (3 X 1) direction.

			a(1) = 1 as the only way to tile a 1 X 1 square is with a square with dimensions 1 X 1.
a(2) = 1 as the only way to tile a 2 X 2 square is with a square with dimensions 2 X 2.
a(3) = 33. The possible tilings, excluding those equivalent by symmetry, are:
.
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
  |   |       |   |   |       |   |       |   |   |           |   |   |       |
  +---+---+---+   +---+---+---+   +---+---+   +   +---+---+---+   +---+---+---+
  |   |       |   |           |   |       |   |   |           |   |       |   |
  +   +       +   +           +   +       +   +   +           +   +       +   +
  |   |       |   |           |   |       |   |   |           |   |       |   |
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
.
The first tiling can occur in 4 different ways, the second in 8 different ways, the third in 8 different ways, the fourth in 4 different ways and the fifth in 8 different ways. There is also the single 3 X 3 rectangle. This gives 33 ways in total.

Cf. A360725 (oblongs), A360499, A360498, A182275, A004003, A099390, A065072.

A360498 Number of ways to tile an n X n square using oblongs with distinct dimensions.

Original entry on oeis.org

0, 0, 4, 12, 256, 3620, 87216, 2444084, 87181220

Offset: 1

Scott R. Shannon, Feb 09 2023

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

			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) = 12. The possible tilings, excluding those equivalent by symmetry, are:
.
  +---+---+---+---+   +---+---+---+---+
  |   |           |   |               |
  +   +           +   +---+---+---+---+
  |   |           |   |               |
  +---+---+---+---+   +               +
  |               |   |               |
  +               +   +               +
  |               |   |               |
  +---+---+---+---+   +---+---+---+---+
.
The first tiling can occur in 8 different way and the second in 4 different ways, giving 12 ways in total.

Cf. A360499 (rectangles), A004003, A099390, A065072, A233320, A230031.

A334089 a(n) = sqrt(A334088(n)/2^(n-1)).

Original entry on oeis.org

1, 2, 13, 272, 18281, 3944920, 2732887529, 6077512159232, 43384923739812577, 994156445200670735008, 73125714588602035608260981, 17265651822746410593596262486016, 13085551252412040683513520733767180041, 31834381760532514451976501491991780699626368

Offset: 1

Seiichi Manyama, Apr 14 2020

nonn

Wikipedia, Chebyshev polynomials
Wikipedia, Resultant

Cf. A065072, A334088, A340295.

Table[Resultant[ChebyshevT[2*n, x/2], ChebyshevT[2*n, I*x/2], x]^(1/4) / 2^((n-1)/2), {n, 1, 15}] (* Vaclav Kotesovec, Apr 14 2020 *)

{a(n) = sqrtint(sqrtint(polresultant(polchebyshev(2*n, 1, x/2), polchebyshev(2*n, 1, I*x/2)))/2^(n-1))}

a(n) ~ exp(2*G*n^2/Pi) / 2^(3*n/2 - 5/8), where G is Catalan's constant A006752. - Vaclav Kotesovec, Apr 14 2020

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.

Showing 1-10 of 14 results. Next

cp's OEIS Frontend

A004003 Number of domino tilings (or dimer coverings) of a 2n X 2n square.

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A007726 Number of spanning trees of quarter Aztec diamonds of order n.

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A256043 Order of all-2s configuration on the 2n X 2n sandpile grid graph.

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A360499 Number of ways to tile an n X n square using rectangles with distinct dimensions.

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A340052 a(n) = Product_{1<=i

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A340185 Number of spanning trees in the halved Aztec diamond HOD_n.

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A360256 Number of ways to tile an n X n square using rectangles with distinct height X width dimensions.

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A360498 Number of ways to tile an n X n square using oblongs with distinct dimensions.

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