|
%I #7 Oct 22 2021 23:47:59
%S 1,1,4,1,3,2,1,14,7,128,1,11,5,71,36,1,52,18,1358,539,43264,1,41,13,
%T 769,281,17753,6728,1,194,47,14852,4271,1452866,434657,151519232,1,
%U 153,34,8449,2245,603126,167089,46069729,12988816,1,724,123,163534,34276,49704772,10894561,16236962114,3625549353,5475450241024
%N 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).
%C 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.
%H Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter and Tianyuan Xu, <a href="https://doi.org/10.37236/4472">Sandpiles and Dominos</a>, El. J. Comb., 22 (2015), P1.66.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Abelian_sandpile_model">Abelian sandpile model</a>
%F T(2m, 2n) = A187617(m, n) = A187618(m, n). [Florescu et al., Theorem 15]
%F T(2m, 2n-1) = T(2n-1, 2m) = A103997(m, n). [Florescu et al., Theorem 18]
%F 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]
%F A256045(m, n) divides T(m, n), T(m, n) divides A116469(m+1, n+1).
%F This triangle can obviously be extended to n > m as T(m, n) = T(n, m).
%e The triangle begins:
%e 1
%e 1 4
%e 1 3 2
%e 1 14 7 128
%e 1 11 5 71 36
%e 1 52 18 1358 539 43264
%e 1 41 13 769 281 17753 6728
%e ...
%e See Fig. 9 of the paper by Florescu et al. for the T(4, 4) = 36 symmetric recurrent sandpiles on a 4x4 grid.
%Y Cf. A348567, A187617, A187618, A103997, A256045, A116469.
%K nonn,tabl
%O 0,3
%A _Andrey Zabolotskiy_, Oct 22 2021
|
