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.)

330, 331, 332, 233, 1301, 1203, 1301, 1302, 1310, 1311, 1302, 1303, 1311, 1312, 1313, 1310, 1311, 1213, 1320, 1321, 1223, 1311, 1312, 1320, 1321, 1322, 1330, 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

Offset: 1

Author

M. F. Hasler, Mar 07 2018

Keywords

• nonn
• fini
• full
• tabl

Comments

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.

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.

Examples

			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).
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.)
A \ B: 0112  0113  0121  0122  0123  0131  0132  0133  0211  ...
0112 :\0330\ 0331  0233  1301  1302  1310  1311  1312  0323  ...
0113 : 0331 \0332\ 1301  1302  1310  1311  1312  1313  1031  ...
0121 : 0233  1301 \1203\ 1310  1311  1213  1320  1321  0332  ...
0122 : 1301  1302  1310 \1311\ 1312  1320  1321  1322  0333  ...
0123 : 1302  1303  1311  1312 \1313\ 1321  1322  1323  2002  ...
0131 : 1310  1311  1213  1320  1321 \1223\ 1330  1331  2011  ...
0132 : 1311  1312  1320  1321  1322  1330 \1331\ 1332  2012  ...
0133 : 1312  1313  1321  1322  1323  1331  1332 \1333\ 2012  ...
0211 : 0323  1031  0332  0333  2002  1303  2011  2012 \1023\ ...
...
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..18528. (Complete sequence: row / column 1..192, flattened.)

Crossrefs

For links, references etc. see the main entry A300006.

Programs

PARI

A300009(m,n)=m2d(spa(S2[m],S2[n])) \\ with m2d(), spa() and S2 defined in A300006.