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 A351379 #35 Mar 05 2022 22:33:51 %S A351379 24,54,288,480,744,1062,1968,2616,3480,4398,6000,7344,9744,11628, %T A351379 14256,16632,20376,23436,27312,30984,37104,41652,47424,52776,60432, %U A351379 66636,75552,82752,93288,101676,112488,121968,135768,146436,163032,175182,191256,204690,221784,236646,257400,273738,296784 %N A351379 The number of grains of sand in the identity element for the 3D sandpile group on an n X n X n cubic grid. %C A351379 The 3D sandpile model follows the same rules as the 2D model except that cells topple and transfer one grain of sand to their six nearest neighbors when the cell contains 6 or more grains. Cells containing 0 to 5 grains are stable. %C A351379 See A307652 for details of the sandpile group identity. %H A351379 Noah Doman, <a href="https://fse.studenttheses.ub.rug.nl/21391/1/bMath_2020_DomanN.pdf">The Identity of the Abelian Sandpile Group</a>, Bachelor Thesis, University of Groningen, January 2020. %H A351379 Luis David Garcia-Puente and Brady Haran, <a href="https://youtu.be/1MtEUErz7Gg">Sandpiles</a>, Numberphile video, YouTube.com, Jan. 13, 2017. %H A351379 Yvan Le Borgne and 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 A351379 Scott R. Shannon, <a href="/A351379/a351379_6.gif">Middle layer of the 100x100x100 identity</a>. This contains 3486864 grains. For this and other images, white=0, red=1, green=2, blue=3, violet=4, yellow=5 grains per cell. %H A351379 Scott R. Shannon, <a href="/A351379/a351379_7.gif">Top layer of the 100x100x100 identity</a>. %H A351379 Scott R. Shannon, <a href="/A351379/a351379_8.gif">Middle layer of the 101x101x101 identity</a>. Similarly to the 2D sandpile model, when n is odd the middle layers have a cross-like pattern. %H A351379 Zach J. Shannon, <a href="/A351379/a351379.png">3D image of the full 80x80x80 identity</a>. The same colors as above are used except cells with no grains are shown as vacancies, not white. %H A351379 Zach J. Shannon, <a href="/A351379/a351379_1.png">3D image of half the 80x80x80 identity</a>. %H A351379 Zach J. Shannon, <a href="/A351379/a351379_2.png">3D image of half the 80x80x80 identity without the cells containing 5 grains</a>. %H A351379 Zach J. Shannon, <a href="/A351379/a351379_3.png">3D image of half the 80x80x80 identity showing only the cells containing 5 grains</a>. %F A351379 Identity element = ([10n] - ([10n])*)* , where [10n] is the all 10's grid of size n X n X n, and (x)* represents the topple stabilization of the grid x. %F A351379 The sequence is approximately fitted by the cubic a(n) ~ 3.48*n^3, where 3.48 corresponds to the approximate grains-per-cube density of the identity element configurations. %e A351379 a(2) = 2 X 2 X 2 grid. Identity: %e A351379 Layer 1: | 3 3 | Layer 2: | 3 3 | %e A351379 | 3 3 | | 3 3 | = 24 grains. %e A351379 a(3) = 3 X 3 X 3 grid. Identity: %e A351379 Layer 1: | 3 2 3 | Layer 2: | 2 1 2 | Layer 3: | 3 2 3 | %e A351379 | 2 1 2 | | 1 0 1 | | 2 1 2 | %e A351379 | 3 2 3 | | 2 1 2 | | 3 2 3 | = 54 grains. %Y A351379 Cf. A307652 (square grid), A259013, A180230, A300006, A007341. %K A351379 nonn %O A351379 2,1 %A A351379 _Scott R. Shannon_ and _Zach J. Shannon_, Feb 09 2022