A348566 Triangle read by rows: T(m, n) is the number of symmetric recurrent sandpiles on an m X n grid (m >= 0, 0 <= n <= m).

1, 1, 4, 1, 3, 2, 1, 14, 7, 128, 1, 11, 5, 71, 36, 1, 52, 18, 1358, 539, 43264, 1, 41, 13, 769, 281, 17753, 6728, 1, 194, 47, 14852, 4271, 1452866, 434657, 151519232, 1, 153, 34, 8449, 2245, 603126, 167089, 46069729, 12988816, 1, 724, 123, 163534, 34276, 49704772, 10894561, 16236962114, 3625549353, 5475450241024

Offset

0,3

Comments

Terms of this triangle count recurrent sandpiles on rectangular grids that have vertical and horizontal symmetries. Terms of A348567 count recurrent sandpiles on square grids that also have diagonal symmetries.

Links

Table of n, a(n) for n=0..54.

Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter and Tianyuan Xu, Sandpiles and Dominos, El. J. Comb., 22 (2015), P1.66.

Wikipedia, Abelian sandpile model

Formula

T(2m, 2n) = A187617(m, n) = A187618(m, n). [Florescu et al., Theorem 15]

T(2m, 2n-1) = T(2n-1, 2m) = A103997(m, n). [Florescu et al., Theorem 18]

T(2m-1, 2n-1) = Product_{h=1..m, k=1..n} 4*(z(h, m) + z(k, n)) where z(k, n) = cos(Pi*(2k-1)/(4n)). [Florescu et al., Theorem 23]

A256045(m, n) divides T(m, n), T(m, n) divides A116469(m+1, n+1).

This triangle can obviously be extended to n > m as T(m, n) = T(n, m).

Example

The triangle begins:
  1
  1  4
  1  3  2
  1 14  7  128
  1 11  5   71  36
  1 52 18 1358 539 43264
  1 41 13  769 281 17753 6728
...
See Fig. 9 of the paper by Florescu et al. for the T(4, 4) = 36 symmetric recurrent sandpiles on a 4x4 grid.

Crossrefs

Cf. A348567, A187617, A187618, A103997, A256045, A116469.

Sequence in context: A368922 A327357 A334810 * A021246 A301907 A197840

Adjacent sequences: A348563 A348564 A348565 * A348567 A348568 A348569

Keyword

nonn,tabl

Author

Andrey Zabolotskiy, Oct 22 2021

Status

approved