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 A300009 #11 Mar 10 2018 19:54:53 %S A300009 330,331,332,233,1301,1203,1301,1302,1310,1311,1302,1303,1311,1312, %T A300009 1313,1310,1311,1213,1320,1321,1223,1311,1312,1320,1321,1322,1330, %U A300009 1331,1312,1313,1321,1322,1323,1331,1332,1333,323,1031,332,333,2002,1303,2011,2012,1023,1031,1032,333,2002,2003,2011,2012,2013,1130 %N A300009 Addition table for the 2 X 2 sandpile group: T(m,n) = A300006(m) (+) A300006(n), for 1 <= m <= n <= 192. (Upper right part of the symmetric matrix.) %C A300009 The sandpile-addition of 2 X 2 matrices is the standard addition, followed by repeated "toppling" of matrix elements > 3, which are decreased by 4 and increase each of their von-Neumann neighbors. A300006 lists all 192 elements of the 2 X 2 sandpile group, the largest subset of the 2 X 2 matrices which forms a group under the sandpile addition, with neutral element e = [2,2;2,2] encoded as A300006(116) = 2222. The symbol (+) denotes sandpile addition indifferently for 2 X 2 matrices and for their decimal encoding. %C A300009 This is the (addition) table of this group, which is abelian, so we list only 1 <= m <= n <= 192, where m, n are the indices of the elements of A300006. %H A300009 M. F. Hasler, <a href="/A300009/b300009.txt">Table of n, a(n) for n = 1..18528</a>. (Complete sequence: row / column 1..192, flattened.) %e A300009 T(1,1) = 0330 represents [0,1;1,2] (+) [0,1;1,2] = [0,3;3,0] (result after "toppling" the plain addition of the first element of A300006 to itself, 0112 + 0112 = 0224). %e A300009 Given that the operation is abelian, the sequence lists only the upper-right (or equivalently, lower left) part of the table: (For reference we mark \abcd\ the diagonal element which is the last one listed of the respective row / column.) %e A300009 A \ B: 0112 0113 0121 0122 0123 0131 0132 0133 0211 ... %e A300009 0112 :\0330\ 0331 0233 1301 1302 1310 1311 1312 0323 ... %e A300009 0113 : 0331 \0332\ 1301 1302 1310 1311 1312 1313 1031 ... %e A300009 0121 : 0233 1301 \1203\ 1310 1311 1213 1320 1321 0332 ... %e A300009 0122 : 1301 1302 1310 \1311\ 1312 1320 1321 1322 0333 ... %e A300009 0123 : 1302 1303 1311 1312 \1313\ 1321 1322 1323 2002 ... %e A300009 0131 : 1310 1311 1213 1320 1321 \1223\ 1330 1331 2011 ... %e A300009 0132 : 1311 1312 1320 1321 1322 1330 \1331\ 1332 2012 ... %e A300009 0133 : 1312 1313 1321 1322 1323 1331 1332 \1333\ 2012 ... %e A300009 0211 : 0323 1031 0332 0333 2002 1303 2011 2012 \1023\ ... %e A300009 ... %o A300009 (PARI) A300009(m,n)=m2d(spa(S2[m],S2[n])) \\ with m2d(), spa() and S2 defined in A300006. %Y A300009 For links, references etc. see the main entry A300006. %K A300009 nonn,fini,full,tabl %O A300009 1,1 %A A300009 _M. F. Hasler_, Mar 07 2018