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A subsequence of A227723, showing all the big equivalence classes that contain Boolean functions related to subgroups of nimber addition (A190939).

Forms a triangle with row lengths A034343 = 1, 1, 2, 4, 8, 16, 36, 80...:

1,

3,

6, 15,

24, 60, 105, 255,

384, 960, 1632, 1680, 4080, 15555, 27030, 65535...

The left column a( 1,2,4,8,16,32,68,148... ) = a( A076766 ) = 3 ,6, 24, 384, 98304... is probably A001146 * 3/2, which is also A006017( A000079 ).

The first A076766(n) entries correspond to the first A006116(n) entries of A190939. (The first 148 here, for n = 7, correspond to the first 29212 there.) The entries of A190939 can be generated from this sequence.

Among the first A076766(n) entries are A076831(n;0...n) with weight 2^0...2^n. (Among the first 148 are 1, 7, 23, 43, 43, 23, 7, 1 with weights 1, 2, 4, 8, 16, 32, 64, 128.)

a(n) appears to be divisible by 3 for n>0, and the odd part of a(n) is almost always squarefree. - Ralf Stephan, Aug 02 2013

This is a subsequence of A227723 containing representatives of the big equivalence classes representing functions which are monotone in each of their variables. Because of the ordering of functions in the original sequence, the functions represented are actually positive functions (i.e., positive in each of their variables).

Two Boolean functions belong to the same small equivalence class (sec) when they can be expressed by each other by negating arguments. E.g., when f(p,~q,r) = g(p,q,r), then f and g belong to the same sec. Geometrically this means that the functions correspond to hypercubes with 2-colored vertices that are equivalent up to reflection (i.e., exchanging opposite hyperfaces).

Boolean functions correspond to integers, so each sec can be denoted by the smallest integer corresponding to one of its functions. There are A000231(n) small equivalence classes of n-ary Boolean functions. Ordered by size they form the finite sequence A_n. It is the beginning of A_(n+1) which leads to this infinite sequence A.

All rows of this array are infinite permutations of the nonnegative integers. Row m (counted from 0) is always generated by modifying the sequence of nonnegative integers in the following way: The sequence of integers is written in reverse binary. Than the finite permutation p_m (row m of array A055089) is applied on the digits of all entries.

The rows of the top left n! X 2^n submatrix describe the rotations and reflections of the n-hypercube that preserve the binary digit sums of the vertex numbers. With permutation composition these permutations form the symmetric group S_n.

Applying such a permutation on the binary string of a Boolean function gives the string of a function in the same big equivalence class (compare A227723).

Triangle row m has length 2^n for m in the interval [(n-1)!,n![. The rest of the array row repeats the same pattern. The first digit of the rest is the digit before plus one.