OFFSET
0,2
COMMENTS
From Andrei Zabolotskii, Dec 01 2025: (Start)
a(n) is also the least common denominator (over all possible starting positions) of the halved expected numbers of random steps to escape an n X n grid. Indeed: a(n) is the smallest integer such that the solution t_n of the linear equation system Laplacian(t_n) = two*a(n) is integral, where "two" is the all-2s vector and t_n is the toppling function that is stabilizing for the sum of the right-hand side and any recurrent sandpile; on the other hand, the vector e_n of the expected numbers of steps to escape satisfies (1/4)*Laplacian(e_n) = one, the all-1s vector.
One component of the vector e_n is A376606(n)/A376607(n), so a(n) is a multiple of the denominator of half that fraction, i.e. of A376607(n) or 2*A376607(n); the latter is the case when A376606(n) is odd, which seems to happen when n is odd. In fact, a(n) = A376607(n) * ((n mod 2) + 1), except for n = 17, 39... (End)
LINKS
Andrei Zabolotskii, Table of n, a(n) for n = 0..44
Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter, and Tianyuan Xu, Sandpiles and Dominos, Electronic Journal of Combinatorics, Volume 22(1), 2015.
PROG
(PARI)
droprob(n, moves=[[1, 0], [0, 1], [0, -1], [-1, 0]]) = {my(np=n^2+1, M=matrix(np), P=1/#moves); for(t=1, #moves, for( i=1, n, my(ti=i+moves[t][1]); for(j=1, n, my(tj=j+moves[t][2]); my(m=(i-1)*n+j); if(ti<1 || ti>n || tj<1 || tj>n, M[m, np]+=P, my(mt=(ti-1)*n+tj); M[m, mt]+=P)))); 1/(matid(np)-M)};
a(n) = {my(M=droprob(n)); denominator(vector(n^2, i, vecsum(M[, i])/2))}; \\ Andrei Zabolotskii, Nov 30 2025, after A376606
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Mar 15 2015
EXTENSIONS
Terms a(0)-a(1) and a(11) onwards from Andrei Zabolotskii, Nov 24 2025, corrected Nov 30 2025
STATUS
approved