A385665 Triangle read by rows: T(n,k) is the number of 2n-bead balanced bicolor necklaces that can be rotated into their complements in k different ways.

1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 3, 0, 0, 0, 1, 5, 1, 1, 0, 0, 1, 9, 0, 0, 0, 0, 0, 1, 16, 2, 0, 1, 0, 0, 0, 1, 28, 0, 1, 0, 0, 0, 0, 0, 1, 51, 3, 0, 0, 1, 0, 0, 0, 0, 1, 93, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 170, 5, 2, 1, 0, 1, 0, 0, 0, 0, 0, 1, 315, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Tilman Piesk, Jul 06 2025

Keywords

• nonn
• tabl

Comments

Let X = A003239, Y = A000013, Z = A000048.

Rotations producing the complementary and the same necklace: CR and SR

There are X(n) balanced bicolor necklaces (BBN) of length 2n. (Central numbers of A047996.)

Y(n) among them are self-complementary (SCBBN). (They can be rotated so that all beads change color.)

Z(n) among those are primitive (not periodic). Each has a unique CR and SR. (SR is trivial rotation.)

The other Y(n)-Z(n) = A115118(n) SCBBN have multiple CR and SR.

T(n,k) SCBBN have k different CR and SR.

Column 1 is Z. The other columns have the same positive entries, each preceded by k-1 zeros.

One could add a column 0 to this triangle, whose entries would be X(n)-Y(n) = 2*A386388(n).

Triangle A385666 does the same for SR of all BBN.

Examples

			Triangle begins:
      k    1  2  3  4  5  6  7  8  9 10 11 12 12 14 15 16     A000013(n)
  n
  1        1  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .             1
  2        1  1  .  .  .  .  .  .  .  .  .  .  .  .  .  .             2
  3        1  .  1  .  .  .  .  .  .  .  .  .  .  .  .  .             2
  4        2  1  .  1  .  .  .  .  .  .  .  .  .  .  .  .             4
  5        3  .  .  .  1  .  .  .  .  .  .  .  .  .  .  .             4
  6        5  1  1  .  .  1  .  .  .  .  .  .  .  .  .  .             8
  7        9  .  .  .  .  .  1  .  .  .  .  .  .  .  .  .            10
  8       16  2  .  1  .  .  .  1  .  .  .  .  .  .  .  .            20
  9       28  .  1  .  .  .  .  .  1  .  .  .  .  .  .  .            30
 10       51  3  .  .  1  .  .  .  .  1  .  .  .  .  .  .            56
 11       93  .  .  .  .  .  .  .  .  .  1  .  .  .  .  .            94
 12      170  5  2  1  .  1  .  .  .  .  .  1  .  .  .  .           180
 13      315  .  .  .  .  .  .  .  .  .  .  .  1  .  .  .           316
 14      585  9  .  .  .  .  1  .  .  .  .  .  .  1  .  .           596
 15     1091  .  3  .  1  .  .  .  .  .  .  .  .  .  1  .          1096
 16     2048 16  .  2  .  .  .  1  .  .  .  .  .  .  .  1          2068
Examples for n=4 with necklaces of length 8:
T(4, 1) = 2 necklaces can be rotated into their complements in k=1 way:
 00001111 can be turned into 11110000 by rotating 4 places to the right.
 00101101 can be turned into 11010010 by rotating 4 places to the right.
T(4, 2) = 1 necklace can be rotated into its complement in k=2 ways:
 00110011 can be turned into 11001100 by rotating 2 or 6 places to the right.
T(4, 4) = 1 necklace can be rotated into its complement in k=4 ways:
 01010101 can be turned into 10101010 by rotating 1, 3, 5 or 7 places to the right.
		

Links

• Tilman Piesk, Rows 1..32, flattened
• Tilman Piesk, Triangle T(n,k)*2*n/k with row sums A045654(n)

Crossrefs

Cf. A003239, A000013, A000048, A385666, A386388, A045654, A115118.

Formula

T(n,k) = A000048(n/k) iff n divisible by k, otherwise 0.