A348566 - OEIS

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

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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: