A355255 Irregular table read by rows: a(n,k) gives the number of distinct necklaces that appear in the following procedure: starting with the n-bead, (0,1)-necklace given by k written in binary, repeatedly take the first differences (mod 2) of the beads. 0 <= k < 2^n.

1, 1, 2, 1, 3, 3, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 5, 5, 4, 5, 3, 4, 5, 5, 4, 3, 5, 4, 5, 5, 2, 1, 4, 4, 3, 4, 3, 3, 4, 4, 3, 3, 4, 3, 4, 4, 3, 4, 3, 3, 4, 3, 4, 4, 3, 3, 4, 4, 3, 4, 3, 3, 2, 1, 4, 4, 3, 4, 2, 3, 3, 4, 2, 2, 4, 3, 4, 3, 2, 4, 2, 2, 4, 2, 3, 4, 3, 3, 4, 4, 1, 3, 3, 2, 4, 4, 3, 2, 3, 2, 4, 4, 2, 2, 4, 3, 3, 4, 1, 3, 4, 3, 3, 4, 2, 4, 3, 1, 4, 3, 2, 3, 4, 2, 4, 4, 2

Offset

0,3

Comments

For j >= 1, the sequence a(j,1) begins

2, 3, 2, 5, 4, 4, 8, 9, 8, 8, 32, 8, 64, 16, 16, 17, 16, 16, 512, 16, 64, 64, 2048, 16, 1024, 128, 512, 32, 16384, 32, ...

Conjecture: a(2^m,1) = 2^m + 1 for all m > 1.

Conjecture: a(m,1) is a power of 2 whenever m is not a power of 2.

The sequence of the number of distinct values in the n-th row begins 1, 2, 3, 2, 5, 4, 4, 4, 9, 4, 8, 4, 8, 4, 10, 6, 17, 6, 10, ... - Peter Kagey, Jul 03 2022

Links

Peter Kagey, Rows n = 0..12 of the triangle, flattened

Example

Table begins:
n\k | 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15
----+-----------------------------------------------
  0 | 1;
  1 | 1, 2;
  2 | 1, 3, 3, 2;
  3 | 1, 2, 2, 1, 2, 1, 1, 2;
  4 | 1, 5, 5, 4, 5, 3, 4, 5, 5, 4, 3, 5, 4, 5, 5, 2;
... | ...
a(5,13) = 4 because 13 is 01101 in binary; the sequence of first differences is 01101, 10111, 11000, 01001, 11011, ...; and 10111 is the same necklace as 11011.

Crossrefs

Cf. A038556, A334594.

Sequence in context: A355686 A308530 A302555 * A336692 A332434 A262209

Adjacent sequences: A355252 A355253 A355254 * A355256 A355257 A355258

Keyword

nonn,tabf,base,look

Author

Peter Kagey, Jun 26 2022

Status

approved