A300007 Index of the inverse/opposite of the n-th 2 X 2 sandpile A300006(n). An involution (self-inverse permutation) of the integers [1..192].

95, 82, 92, 38, 37, 47, 35, 34, 83, 68, 67, 81, 30, 29, 28, 36, 26, 25, 24, 69, 53, 52, 66, 19, 18, 17, 27, 15, 14, 13, 54, 40, 51, 8, 7, 16, 5, 4, 43, 32, 191, 190, 39, 188, 187, 186, 6, 184, 183, 182, 33, 22, 21, 31, 177, 176, 175, 189, 173, 172, 171, 185, 169, 168

Offset: 1

Author

M. F. Hasler, Mar 07 2018

Keywords

• nonn
• fini
• full

Comments

See A300006 for the definition of the sandpile addition of 2 X 2 matrices.

A300006(116) = 2222 represents the neutral element e = [2,2;2,2], so a(116) = 116.

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).

Examples

			a(1) = 95 because A300006(1) + A300006(95) = 0112 + 2110 = 2222 represents the unit element of the 2 X 2 sandpile group.
a(2) = 82 because A300006(2) + A300006(82) = 0113 + 2003 = 2116 "topples" to 2222.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..192

Crossrefs

Cf. A300006, A300008, A300009.

Programs

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.