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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.) 4
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

LINKS

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

EXAMPLE

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\ ...

...

PROG

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

CROSSREFS

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

Sequence in context: A190920 A145274 A174848 * A244264 A117345 A237189

Adjacent sequences:  A300006 A300007 A300008 * A300010 A300011 A300012

KEYWORD

nonn,fini,full,tabl

AUTHOR

M. F. Hasler, Mar 07 2018

STATUS

approved

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Last modified November 28 16:42 EST 2021. Contains 349413 sequences. (Running on oeis4.)