%I
%S 95,82,92,38,37,47,35,34,83,68,67,81,30,29,28,36,26,25,24,69,53,52,66,
%T 19,18,17,27,15,14,13,54,40,51,8,7,16,5,4,43,32,191,190,39,188,187,
%U 186,6,184,183,182,33,22,21,31,177,176,175,189,173,172,171,185,169,168
%N Index of the inverse/opposite of the nth 2 X 2 sandpile A300006(n). An involution (selfinverse permutation) of the integers [1..192].
%C See A300006 for the definition of the sandpile addition of 2 X 2 matrices.
%C A300006(116) = 2222 represents the neutral element e = [2,2;2,2], so a(116) = 116.
%C A300006 forms a group for the sandpile addition, so for each A300006(n) there exists a unique m, given here as a(n), such that A300006(n) (+) A300006(m) = 2222 (where (+) means the sandpile addition). A300006(m) is also listed as A300008(n).
%H M. F. Hasler, <a href="/A300007/b300007.txt">Table of n, a(n) for n = 1..192</a>
%e a(1) = 95 because A300006(1) + A300006(95) = 0112 + 2110 = 2222 represents the unit element of the 2 X 2 sandpile group.
%e a(2) = 82 because A300006(2) + A300006(82) = 0113 + 2003 = 2116 "topples" to 2222.
%o (PARI) A300007(n)=for(m=1,#S2,spa(S2[n],S2[m])==[2,2;2,2]&&return(m)) \\ S2 and spa() being defined as in A300006.
%Y Cf. A300006, A300008, A300009.
%K nonn,fini,full
%O 1,1
%A _M. F. Hasler_, Mar 07 2018
