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

Author

Peter Kagey, Jun 26 2022

Keywords

• nonn
• tabf
• base
• look

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A038556, A334594.