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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.

%I A349991 #9 Dec 18 2021 15:01:00
%S A349991 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 A349991 7,7,18,19,19,19,19,19,18,19,19,19,19,19,18,19,19,19,19,19,18,19,19,
%U A349991 19,19,19,18,19,19,19,19,19,18,19,19,19,19,19,24,25
%N 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.
%C A349991 A site is unstable when it holds 6 or more grains.
%C A349991 As long as there is an unstable site:
%C A349991 - choose such an unstable site,
%C A349991 - remove 6 grains from this site and add 1 grain to each of its six neighbors.
%C A349991 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 A349991 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 A349991 Rémy Sigrist, <a href="/A349991/a349991.txt">C++ program for A349991</a>
%H A349991 Wikipedia, <a href="https://en.wikipedia.org/wiki/Abelian_sandpile_model#Sandpile_models_on_infinite_grids">Sandpile models on infinite grids</a>
%F A349991 a(6*n) + 1 = a(6*n + k) for k = 1..5.
%e A349991 For n = 54:
%e A349991 - after stabilization, we have the following configuration:
%e A349991              2
%e A349991           4     4
%e A349991        2     3     2
%e A349991           3     3
%e A349991        4           4
%e A349991           3     3
%e A349991        2     3     2
%e A349991           4     4
%e A349991              2
%e A349991 - we have 18 nonnempty sites,
%e A349991 - so a(54) = 18.
%o A349991 (C++) See Links section.
%Y A349991 Cf. A349990.
%K A349991 nonn
%O A349991 0,7
%A A349991 _Rémy Sigrist_, Dec 08 2021