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%I #11 Mar 10 2018 19:54:53
%S 330,331,332,233,1301,1203,1301,1302,1310,1311,1302,1303,1311,1312,
%T 1313,1310,1311,1213,1320,1321,1223,1311,1312,1320,1321,1322,1330,
%U 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 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 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 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 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 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 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 A \ B: 0112 0113 0121 0122 0123 0131 0132 0133 0211 ...
%e 0112 :\0330\ 0331 0233 1301 1302 1310 1311 1312 0323 ...
%e 0113 : 0331 \0332\ 1301 1302 1310 1311 1312 1313 1031 ...
%e 0121 : 0233 1301 \1203\ 1310 1311 1213 1320 1321 0332 ...
%e 0122 : 1301 1302 1310 \1311\ 1312 1320 1321 1322 0333 ...
%e 0123 : 1302 1303 1311 1312 \1313\ 1321 1322 1323 2002 ...
%e 0131 : 1310 1311 1213 1320 1321 \1223\ 1330 1331 2011 ...
%e 0132 : 1311 1312 1320 1321 1322 1330 \1331\ 1332 2012 ...
%e 0133 : 1312 1313 1321 1322 1323 1331 1332 \1333\ 2012 ...
%e 0211 : 0323 1031 0332 0333 2002 1303 2011 2012 \1023\ ...
%e ...
%o (PARI) A300009(m,n)=m2d(spa(S2[m],S2[n])) \\ with m2d(), spa() and S2 defined in A300006.
%Y For links, references etc. see the main entry A300006.
%K nonn,fini,full,tabl
%O 1,1
%A _M. F. Hasler_, Mar 07 2018
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