A300009 - OEIS

%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