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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
KEYWORD
nonn,fini,full,tabl
AUTHOR
M. F. Hasler, Mar 07 2018
STATUS
approved

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Last modified March 29 08:13 EDT 2024. Contains 371265 sequences. (Running on oeis4.)