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 A349991 For any n >= 0, consider a sandpile model on the infinite hexagonal lattice starting with n grains at the origin, the other sites being empty; a(n) gives the number of nonempty sites after stabilization of this sandpile model. 2

%I #9 Dec 18 2021 15:01:00

%S 0,1,1,1,1,1,6,7,7,7,7,7,6,7,7,7,7,7,6,7,7,7,7,7,6,7,7,7,7,7,6,7,7,7,

%T 7,7,18,19,19,19,19,19,18,19,19,19,19,19,18,19,19,19,19,19,18,19,19,

%U 19,19,19,18,19,19,19,19,19,18,19,19,19,19,19,24,25

%N For any n >= 0, consider a sandpile model on the infinite hexagonal lattice starting with n grains at the origin, the other sites being empty; a(n) gives the number of nonempty sites after stabilization of this sandpile model.

%C A site is unstable when it holds 6 or more grains.

%C As long as there is an unstable site:

%C - choose such an unstable site,

%C - remove 6 grains from this site and add 1 grain to each of its six neighbors.

%C This procedure is guaranteed to result in a stable configuration, which does not depend on the order in which we treat the unstable sites.

%H Rémy Sigrist, <a href="/A349991/a349991.png">Colored representation of the stabilized configuration for n = 1000000</a> (white, green, purple, gold, blue and red pixels correspond to sites with 0, 1, 2, 3, 4 and 5 grains, respectively)

%H Rémy Sigrist, <a href="/A349991/a349991.txt">C++ program for A349991</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Abelian_sandpile_model#Sandpile_models_on_infinite_grids">Sandpile models on infinite grids</a>

%F a(6*n) + 1 = a(6*n + k) for k = 1..5.

%e For n = 54:

%e - after stabilization, we have the following configuration:

%e 2

%e 4 4

%e 2 3 2

%e 3 3

%e 4 4

%e 3 3

%e 2 3 2

%e 4 4

%e 2

%e - we have 18 nonnempty sites,

%e - so a(54) = 18.

%Y Cf. A349990.

%K nonn

%O 0,7

%A _Rémy Sigrist_, Dec 08 2021

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