bibkey: arndt2017a293452 authors: Joerg Arndt year: 2017 title: “OEIS A293452, Triangle T(n,k) read by rows: T(n,k) is the number of iterations to reach a final state for an n X k lattice of sandpiles on a torus according to rules specified in A249872” doi: null url: https://oeis.org/A293452 claim: “The entry tabulates the number of toppling iterations of the sandpile of A249872 on the n X k torus and conjectures T(n,1) = A023855(n).” strata_touched:

• D5/S3/StatisticalMechanics/Sandpiles/TorusColumnToppling license: citation-only triage: anchor

OEIS A293452

A293452 (Joerg Arndt, 2017) is the triangle

T(n,k) is the number of iterations to reach a final state for an n X k lattice of sandpiles on a torus according to rules specified in A249872.

with the FORMULA lines

T(n,n) = A249872(n).

Conjecture: T(n,1) = A023855(n).

The rules are those of A249872 (Lars Blomberg, 2014):

Let the lattice be c[i,j], 0 <= i,j < n. Fill each cell except c[0,0] with 4 grains of sand. Until all c[i,j] < 4, do the following: Find a c[i,j] >= 4. (According to Knuth, it does not matter which cell is chosen, the result will be the same.) Decrement the chosen cell by 4 and increment its 4 neighbors by 1. c[0,0] is never increased, sand grains placed here are lost. The number of iterations needed is a(n).

A023855 (offset 1) is

a(n) = 1*(n) + 2*(n-1) + 3*(n-2) + … + (n+1-k)*k, where k = floor((n+1)/2).

The column T(n,1) of the entry’s data is 0, 1, 2, 7, 10, 22, 28, 50, 60, 95 for n = 1, …, 10, while A023855 begins 1, 2, 7, 10, 22, 28, 50, 60, 95, 110, so the data match A023855(n − 1).

Verified locator

• URL: https://oeis.org/A293452
• Version and location: OEIS A293452, revision 19 (2022-03-11), FORMULA; entry by Joerg Arndt, 2017-10-09.
• Related entries: https://oeis.org/A249872 (revision 29), https://oeis.org/A023855 (revision 89).