cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Original entry on oeis.org

1, 2, 36, 6728, 12988816, 258584046368, 53060477521960000, 112202208776036178000000, 2444888770250892795802079170816, 548943583215388338077567813208427340288, 1269984011256235834242602753102293934298576249856
Offset: 0

Views

Author

N. J. A. Sloane, Greg Huber

Keywords

Comments

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

Examples

			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)}

References

  • 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.

Links

Crossrefs

Cf. A028420, A006253, A006125, A065072, A124997, A239273, A256043.
Main diagonal of array A099390 or A187596.

Programs

  • Maple
    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;
  • Mathematica
    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 *)
  • PARI
    {a(n) = sqrtint(polresultant(polchebyshev(2*n, 2, x/2), polchebyshev(2*n, 2, I*x/2)))} \\ Seiichi Manyama, Apr 13 2020
  • Python
    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

Formula

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

Extensions

Corrected and extended by David Radcliffe

A007341 Number of spanning trees in n X n grid.

Original entry on oeis.org

1, 4, 192, 100352, 557568000, 32565539635200, 19872369301840986112, 126231322912498539682594816, 8326627661691818545121844900397056, 5694319004079097795957215725765328371712000, 40325021721404118513276859513497679249183623593590784, 2954540993952788006228764987084443226815814190099484786032640000
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Kreweras calls this the complexity of the n X n grid.
a(n) is the number of perfect mazes made from a grid of n X n cells. - Leroy Quet, Sep 08 2007
Also number of domino tilings of the (2n-1) X (2n-1) square with upper left corner removed. For n=2 the 4 domino tilings of the 3 X 3 square with upper left corner removed are:
. ._. . ._. . ._. . ._.
.|__| .|__| .| | | .|___|
| |_| | | | | | ||| |_| |
||__| |||_| ||__| |_|_| - Alois P. Heinz, Apr 15 2011
Indeed, more is true. Let L denote the (2*n - 1) X (2*n - 1) square lattice graph with vertices (i,j), 1 <= i,j <= 2*n-1. Call a vertex (i,j) odd if both coordinates i and j are odd. Then there is a bijection between the set of spanning trees on the square n X n grid and the set of domino tilings of L with an odd boundary point removed. See Tzeng and Wu, 2002. This is a divisibility sequence, i.e., if n divides m then a(n) divides a(m). - Peter Bala, Apr 29 2014
Also, a(n) is the order of the sandpile group of the (n-1)X(n-1) grid graph. This is because the n X n grid is dual to (n-1)X(n-1) grid + sink vertex, and the latter is related to the sandpiles by the burning bijection. See Járai, Sec. 4.1, or Redig, Sec. 2.2. In M. F. Hasler's comment below, index n refers to the size of the grid underlying the sandpile. - Andrey Zabolotskiy, Mar 27 2018
From M. F. Hasler, Mar 07 2018: (Start)
The sandpile addition (+) of two n X n matrices is defined as the ordinary addition, followed by the topple-process in which each element larger than 3 is decreased by 4 and each of its von Neumann neighbors is increased by 1.
For any n, there is a neutral element e_n such that the set S(n) = { A in M_n({0..3}) | A (+) e_n = A } of matrices invariant under sandpile addition of e_n, forms a group, i.e., each element A in S(n) has an inverse A' in S(n) such that A (+) A' = e_n. (For n > 1, e_n cannot be the zero matrix O_n, because for this choice S(n) would include, e.g., the all 1's matrix 1_n which cannot have an inverse X such that 1_n (+) X = O_n. The element e_n is the unique nonzero matrix such that e_n (+) e_n = e_n.)
The present sequence lists the size of the abelian group (S(n), (+), e_n). See the example section for the e_n. The elements of S(2) are listed as A300006 and their inverses are listed as A300007. (End)

Examples

			From _M. F. Hasler_, Mar 07 2018: (Start)
For n = 1, there exists only one 0 X 0 matrix, e_0 = []; it is the neutral element of the singleton group S(0) = {[]}.
For n = 2, the sandpile addition is isomorphic to addition in Z/4Z, the neutral element is e_1 = [0] and we get the group S(1) isomorphic to (Z/4Z, +).
For n = 3, one finds that e_2 = [2,2;2,2] is the neutral element of the sandpile addition restricted to S(2), having 192 elements, listed in A300006.
For n = 4, one finds that e_3 = [2,1,2;1,0,1;2,1,2] is the neutral element of the sandpile addition restricted to S(3), having 100352 elements.
For n = 5, the neutral element is e_4 = [2,3,3,2; 3,2,2,3; 3,2,2,3; 2,3,3,2]. (End)

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Main diagonal of A116469.
Cf. A300006, A300007, A300008, A300009; A256043, A256045.
Cf. A080690 (number of acyclic orientations), A080691 (number of spanning forests), A349718 (number of spanning trees, reduced for symmetry).

Programs

  • Maple
    a:= n-> round(evalf(2^(n^2-1) /n^2 *mul(mul(`if`(j<>0 or k<>0, 2 -cos(Pi*j/n) -cos(Pi*k/n), 1), k=0..n-1), j=0..n-1), 15 +n*(n+1)/2)): seq(a(n), n=1..20);  # Alois P. Heinz, Apr 15 2011
# uses expression as a resultant
seq(resultant(simplify(ChebyshevU(n-1, x/2)), simplify(ChebyshevU(n-1, (4-x)/2)), x), n = 1 .. 24); # Peter Bala, Apr 29 2014
  • Mathematica
    Table[2^((n-1)^2) Product[(2 - Cos[Pi i/n] - Cos[Pi j/n]), {i, 1, n-1}, {j, 1, n-1}], {n, 12}] // Round
Table[Resultant[ChebyshevU[n-1, x/2], ChebyshevU[n-1, (4-x)/2], x], {n, 1, 12}] (* Vaclav Kotesovec, Apr 15 2020 *)
  • PARI
    {a(n) = polresultant( polchebyshev(n-1, 2, x/2), polchebyshev(n-1, 2, (4-x)/2) )}; /* Michael Somos, Aug 12 2017 */

Formula

a(n) = 2^(n^2-1) / n^2 * product_{n1=0..n-1, n2=0..n-1, n1 and n2 not both 0} (2 - cos(Pi*n1/n) - cos(Pi*n2/n) ). - Sharon Sela (sharonsela(AT)hotmail.com), Jun 04 2002
Equivalently, a(n) = Resultant( U(n-1,x/2), U(n-1,(4-x)/2) ), where U(n,x) is a Chebyshev polynomial of the second kind. - Peter Bala, Apr 29 2014
From Vaclav Kotesovec, Dec 30 2020: (Start)
a(n) ~ 2^(1/4) * Gamma(1/4) * exp(4*G*n^2/Pi) / (Pi^(3/4)*sqrt(n)*(1+sqrt(2))^(2*n)), where G is Catalan's constant A006752.
a(n) = n * 2^(n-1) * A007726(n)^2. (End)

Extensions

More terms and better description from Roberto E. Martinez II, Jan 07 2002

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).

Original entry on oeis.org

1, 1, 3, 29, 901, 89893, 28793575, 29607089625, 97725875584681, 1035449388414303593, 35216739783694029601963, 3844747107219467355553841461, 1347358497824862447450096142795629, 1515633798331963142551890627742773295309
Offset: 0

Views

Author

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

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A099390, A004003, A256043.

Programs

  • Mathematica
    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 *)

Formula

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

Extensions

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

A256045 Triangle read by rows: order of all-2s configuration on the n X k sandpile grid graph.

Original entry on oeis.org

2, 3, 1, 7, 7, 8, 11, 5, 71, 3, 26, 9, 679, 77, 52, 41, 13, 769, 281, 17753, 29, 97, 47, 3713, 4271, 726433, 434657, 272, 153, 17, 8449, 2245, 33507, 167089, 46069729, 901, 362, 123, 81767, 8569, 24852386, 265721, 8118481057, 190818387, 73124, 571, 89, 93127, 18061, 20721019, 4213133, 4974089647, 1031151241, 1234496016491, 89893
Offset: 1

Views

Author

N. J. A. Sloane, Mar 15 2015

Keywords

Examples

			Triangle begins:
[2]
[3, 1]
[7, 7, 8]
[11, 5, 71, 3]
[26, 9, 679, 77, 52]
[41, 13, 769, 281, 17753, 29]
[97, 47, 3713, 4271, 726433, 434657, 272]
[153, 17, 8449, 2245, 33507, 167089, 46069729, 901]
[362, 123, 81767, 8569, 24852386, 265721, 8118481057, 190818387, 73124]
[571, 89, 93127, 18061, 20721019, 4213133, 4974089647, 1031151241, 1234496016491, 89893]
...

Links

  • Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter and Tianyuan Xu, Sandpiles and Dominos, Electronic Journal of Combinatorics, Volume 22(1), 2015.
  • David Perkinson, Lecture 15: Sandpiles, PCMI 2008 Undergraduate Summer School.

Crossrefs

Main diagonal gives A256046, A256043, and A256047.
Cf. A005246, A195549, A348566.

Formula

From Andrey Zabolotskiy, Oct 22 2021: (Start)
It seems that T(k, 1) = A005246(k+2).
For the formula for T(k, 2), see the last theorem of Morar and Perkinson in Perkinson's slides. In particular, T(2*k, 2) = A195549(k).
T(n, k) divides A348566(n, k). (End)

Extensions

Column 1 added by Andrey Zabolotskiy, Oct 22 2021

A256046 Main diagonal of A256045: order of all-2s configuration on the n X n sandpile grid graph.

Original entry on oeis.org

1, 8, 3, 52, 29, 272, 901, 73124, 89893
Offset: 2

Views

Author

N. J. A. Sloane, Mar 15 2015

Keywords

Links

  • Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter, and Tianyuan Xu, Sandpiles and Dominos, Electronic Journal of Combinatorics, Volume 22(1), 2015.

Crossrefs

Cf. A256043, A256045, A256047.

A256047 a(n) = A256046(2n+1): order of all-2s configuration on the (2n+1) X (2n+1) sandpile grid graph.

Original entry on oeis.org

8, 52, 272, 73124
Offset: 1

Views

Author

N. J. A. Sloane, Mar 15 2015

Keywords

Links

  • Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter, and Tianyuan Xu, Sandpiles and Dominos, Electronic Journal of Combinatorics, Volume 22(1), 2015.

Crossrefs

C. A256045, A256046, A256043.

Extensions

Definition corrected by M. F. Hasler, Mar 08 2018
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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