This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A307652 #63 Feb 12 2021 12:16:00
%S A307652 8,12,40,52,72,88,136,160,216,244,320,356,408,448,544,592,704,756,888,
%T A307652 948,1088,1156,1304,1376,1504,1584,1736,1820,1984,2076,2288,2384,2536,
%U A307652 2640,2912,3024,3200,3316,3624,3748,3976,4104,4392,4528,4824,4968,5216,5364,5664,5820,6088,6248,6616
%N A307652 The number of grains of sand in the identity element for the sandpile group on an (n+1) X (n+1) square grid.
%C A307652 The Abelian sandpile model considers the behavior of grains of sand on a square grid when a square topples sand to its nearest neighbors when the number of grains in the square is greater than or equal to 4. Squares on the edge of the board lose sand from the grid when toppling thus a stable configuration for the grid will always occur after a finite number of topples. Starting with the maximal stable grid consisting of 3 grains of sand in all squares, adding sand to one or more squares, and then performing topple stabilization results in a set of recurrent configurations which form the elements of the sandpile group for the given grid size. This group includes one configuration which acts as the identity element for the group, i.e., adding the identity configuration to any chosen group element and then performing topple stabilization results in the chosen group element.
%C A307652 This sequence {a(n)} is the number of sand grains in the identity element of the sandpile group on a square grid of size (n+1) X (n+1).
%H A307652 Scott R. Shannon, <a href="/A307652/b307652.txt">Table of n, a(n) for n = 1..500</a>
%H A307652 Yvan Le Borgne, Dominique Rossin, <a href="https://doi.org/10.1016/S0012-365X(02)00347-3">On the identity of the sandpile group</a>. Discrete Mathematics, 256 (2002) 775-790.
%H A307652 Luis David Garcia-Puente and Brady Haran, <a href="https://youtu.be/1MtEUErz7Gg">Sandpiles</a>, Numberphile video, YouTube.com, Jan. 13, 2017.
%H A307652 Alexander E. Holroyd et al., <a href="https://arxiv.org/abs/0801.3306">Chip-firing and Rotor-Routing on Directed Graphs</a>. arXiv:0801.3306v4 [math.CO], 2013.
%H A307652 Scott R. Shannon, <a href="/A307652/a307652.png">Identity for the 50x50 grid</a>. For this, and other images, black=0, yellow=1, blue=2, red=3 grains.
%H A307652 Scott R. Shannon, <a href="/A307652/a307652_1.png">Identity for the 51x51 grid</a>. This shows the crossed line pattern through the center of the grid which is typical of grids with odd numbered side lengths.
%H A307652 Scott R. Shannon, <a href="/A307652/a307652_2.png">Identity for the 1000x1000 grid</a>.
%H A307652 Scott R. Shannon, <a href="/A307652/a307652_3.png">Identity for the 4000x4000 grid</a>. This contains 37246680 grains.
%H A307652 Scott R. Shannon, <a href="/A307652/a307652_1.java.txt">Simplified Java code for finding the identity element and the sequence a(n)</a>.
%H A307652 Wikipedia, <a href="https://en.wikipedia.org/wiki/Abelian_sandpile_model">Abelian Sandpile Model</a>.
%F A307652 Identity element = ([6n] - ([6n])*)* , where [6n] is the all 6's grid of size (n+1) X (n+1), and (x)* represents the topple stabilization of the grid x.
%F A307652 The sequence is closely fitted by the quadratic a(n) ~ 2.32*n^2, where 2.32 corresponds to the approximate grains per square density of the identity element configurations.
%e A307652 a(1) = 2 X 2 grid.
%e A307652 Identity: | 2 2 |
%e A307652 | 2 2 | = 8 grains.
%e A307652 a(2) = 3 X 3 grid.
%e A307652 Identity: | 2 1 2 |
%e A307652 | 1 0 1 |
%e A307652 | 2 1 2 | = 12 grains.
%e A307652 a(3) = 4 X 4 grid.
%e A307652 Identity: | 2 3 3 2 |
%e A307652 | 3 2 2 3 |
%e A307652 | 3 2 2 3 |
%e A307652 | 2 3 3 2 | = 40 grains.
%e A307652 a(4) = 5 X 5 grid.
%e A307652 Identity: | 2 3 2 3 2 |
%e A307652 | 3 2 1 2 3 |
%e A307652 | 2 1 0 1 2 |
%e A307652 | 3 2 1 2 3 |
%e A307652 | 2 3 2 3 2 | = 52 grains.
%Y A307652 Cf. A259013, A180230, A300006.
%Y A307652 Cf. A007341 (order of the sandpile group of the (n-1)X(n-1) grid graph).
%K A307652 nonn
%O A307652 1,1
%A A307652 _Scott R. Shannon_, Apr 20 2019