A348566 -id:A348566 - cp's OEIS frontend

A103997 Square array T(M,N) read by antidiagonals: number of dimer tilings of a 2M X 2N Moebius strip.

1, 1, 1, 1, 3, 1, 1, 11, 7, 1, 1, 41, 71, 18, 1, 1, 153, 769, 539, 47, 1, 1, 571, 8449, 17753, 4271, 123, 1, 1, 2131, 93127, 603126, 434657, 34276, 322, 1, 1, 7953, 1027207, 20721019, 46069729, 10894561, 276119, 843, 1, 1, 29681, 11332097, 714790675, 4974089647, 3625549353, 275770321, 2226851, 2207, 1

Offset: 0

Ralf Stephan, Feb 26 2005

			Array begins:
  1,   1,     1,        1,          1,             1,               1,
  1,   3,     7,       18,         47,           123,             322,
  1,  11,    71,      539,       4271,         34276,          276119,
  1,  41,   769,    17753,     434657,      10894561,       275770321,
  1, 153,  8449,   603126,   46069729,    3625549353,    289625349454,
  1, 571, 93127, 20721019, 4974089647, 1234496016491, 312007855309063,
  ...

Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter and Tianyuan Xu, Sandpiles and Dominos, El. J. Comb., 22 (2015), P1.66. See Theorem 18.
W. T. Lu and F. Y. Wu, Dimer statistics on the Moebius strip and the Klein bottle, arXiv:cond-mat/9906154 [cond-mat.stat-mech], 1999.
Index entries for sequences related to dominoes

Rows include A005248, A103998.
Columns 1..7 give A001835(n+1), A334135, A334179, A334180, A334181, A334182, A334183.
Main diagonal gives A334124.
Cf. A099390, A103999, A341741, A334178, A256045, A348566.

T[M_, N_] := Product[4Sin[(4n-1)Pi/(4N)]^2 + 4Cos[m Pi/(2M+1)]^2, {n, 1, N}, {m, 1, M}];
Table[T[M - N, N] // Round, {M, 0, 9}, {N, 0, M}] // Flatten (* Jean-François Alcover, Dec 03 2018 *)

T(M, N) = Product_{m=1..M} (Product_{n=1..N} 4*sin(Pi*(4*n-1)/(4*N))^2 + 4*cos(Pi*m/(2*M + 1))^2).

For k > 0, T(n,k) = 2^n * sqrt(Resultant(U_{2*n}(x/2), T_{2*k}(i*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, Apr 15 2020

A187617 Array T(m,n) read by antidiagonals: number of domino tilings of the 2m X 2n grid (m>=0, n>=0).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 13, 36, 13, 1, 1, 34, 281, 281, 34, 1, 1, 89, 2245, 6728, 2245, 89, 1, 1, 233, 18061, 167089, 167089, 18061, 233, 1, 1, 610, 145601, 4213133, 12988816, 4213133, 145601, 610, 1

Offset: 0

N. J. A. Sloane, Mar 11 2011

A099390 is the main entry for this problem.
The even-indexed rows and columns of the square array in A187596.
Row (and column) 2 is given by A122367. - Nathaniel Johnston, Mar 22 2011

			The array begins:
  1,  1,     1,       1,          1,            1, ...
  1,  2,     5,      13,         34,           89, ...
  1,  5,    36,     281,       2245,        18061, ...
  1, 13,   281,    6728,     167089,      4213133, ...
  1, 34,  2245,  167089,   12988816,   1031151241, ...
  1, 89, 18061, 4213133, 1031151241, 258584046368, ...

Alois P. Heinz, Antidiagonals n = 0..26, flattened
N. Allegra, Exact solution of the 2d dimer model: Corner free energy, correlation functions and combinatorics, arXiv:1410.4131 [cond-mat.stat-mech], 2014. See Table 1.
Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter and Tianyuan Xu, Sandpiles and Dominos, El. J. Comb., 22 (2015), P1.66. See Theorem 15.
Index entries for sequences related to dominoes

A187618 is the triangle version.
Cf. A187596, A099390, A348566.
Main diagonal is A004003. Second and third rows give A001519, A188899.

ft:=(m,n)->
2^(m*n/2)*mul( mul(
(cos(Pi*i/(n+1))^2+cos(Pi*j/(m+1))^2), j=1..m/2), i=1..n/2);
T:=(m,n)->round(evalf(ft(m,n),300));

T[m_, n_] := Product[2(2 + Cos[(2j Pi)/(2m+1)] + Cos[(2k Pi)/(2n+1)]), {j, 1, m}, {k, 1, n}];
Table[T[m-n, n] // Round, {m, 0, 8}, {n, 0, m}] // Flatten (* Jean-François Alcover, Aug 05 2018 *)

default(realprecision, 120);
{T(n, k) = round(prod(a=1, n, prod(b=1, k, 4*cos(a*Pi/(2*n+1))^2+4*cos(b*Pi/(2*k+1))^2)))} \\ Seiichi Manyama, Jan 09 2021

More terms from Nathaniel Johnston, Mar 22 2011

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

N. J. A. Sloane, Mar 15 2015

			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]
...

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.

Main diagonal gives A256046, A256043, and A256047.

Cf. A005246, A195549, A348566.

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)

Column 1 added by Andrey Zabolotskiy, Oct 22 2021

A348567 Number of fully symmetric recurrent sandpiles on an n X n grid.

Original entry on oeis.org

1, 4, 2, 32, 12, 832, 232, 69632, 14416, 18719744, 2876576

Offset: 0

Andrey Zabolotskiy, Oct 22 2021

			a(0) = 1, the empty sandpile on an empty grid.
a(1) = 4, all 4 sandpiles on a 1 X 1 grid are fully symmetric.
a(2) = 2 fully symmetric recurrent sandpiles on an 2 X 2 grid are the all-2 and all-3 sandpiles.
Among A348566(4, 4) = 36 sandpiles with horizontal and vertical symmetry, only a(4) = 12 also have diagonal symmetries:
  3333  1331  1331  3223  3223  3333  2222  0330  2332  2332  0330  2222
  3333  3223  3333  2222  2332  3223  2222  3223  3223  3333  3333  2332
  3333  3223  3333  2222  2332  3223  2222  3223  3223  3333  3333  2332
  3333  1331  1331  3223  3223  3333  2222  0330  2332  2332  0330  2222

Cf. A348566, A256046.

A256046(n) divides a(n), a(n) divides A348566(n, n).

cp's OEIS Frontend

A103997 Square array T(M,N) read by antidiagonals: number of dimer tilings of a 2M X 2N Moebius strip.

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A187617 Array T(m,n) read by antidiagonals: number of domino tilings of the 2m X 2n grid (m>=0, n>=0).

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

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A348567 Number of fully symmetric recurrent sandpiles on an n X n grid.

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cp's OEIS Frontend

A103997 Square array T(M,N) read by antidiagonals: number of dimer tilings of a 2*M X 2*N Moebius strip.

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A187617 Array T(m,n) read by antidiagonals: number of domino tilings of the 2m X 2n grid (m>=0, n>=0).

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Maple

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

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Extensions

A348567 Number of fully symmetric recurrent sandpiles on an n X n grid.

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A103997 Square array T(M,N) read by antidiagonals: number of dimer tilings of a 2M X 2N Moebius strip.