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
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-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]
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
KEYWORD
nonn,tabl
AUTHOR
Andrey Zabolotskiy, Oct 22 2021
STATUS
approved