A386388 -id:A386388 - cp's OEIS frontend

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

Tilman Piesk, Jul 06 2025

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.

			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.

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

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

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

A383904 a(n) is the number of complement pairs of primitive 2n-bead balanced binary necklaces.

Original entry on oeis.org

0, 0, 0, 1, 3, 11, 35, 118, 392, 1336, 4587, 15986, 56231, 199854, 716014, 2584742, 9390656, 34315811, 126039218, 465062362, 1723066193, 6407806833, 23910159818, 89493721076, 335912335304, 1264105728831, 4768446686910, 18027215660947, 68291877609003

Offset: 0

Tilman Piesk, Aug 07 2025

A022553(n) is the number of primitive 2n-bead balanced binary necklaces (corresponding to Lyndon words), and A000048 is the number of those that are self-complementary (i.e., can be rotated so that all beads change color). Their difference 2*a(n) is the number of those that are not self-complementary. a(n) is the number pairs of distinct complements.
Doubled entries: 0, 0, 0, 2, 6, 22, 70, 236, 784, 2672, 9174, 31972, 112462, 399708, 1432028, ...
Sequences counting 2n-bead balanced binary necklaces:
                       primitive  imprimitive
                     +-----------------------+---------+
  self-complementary |  A000048     A115118  | A000013 |
   complement pairs  |   this       A387130  | A386388 |
                     +-----------------------+---------+
                     |  A022553     A386946  | A003239 |
                     +-----------------------+---------+

			  n | A022553(n) A000048(n) | 2*a(n)    a(n)
  0 |         1          1  |     0       0
  1 |         1          1  |     0       0
  2 |         1          1  |     0       0
  3 |         3          1  |     2       1
  4 |         8          2  |     6       3
  5 |        25          3  |    22      11
  6 |        75          5  |    70      35
  7 |       245          9  |   236     118
  8 |       800         16  |   784     392
  9 |      2700         28  |  2672    1336
 10 |      9225         51  |  9174    4587
Examples for n=5 with necklaces of length 10:
The total number of necklaces is A003239(5) = 26.
Only A386946(5) = 1 of them is periodic, namely 0101010101.
The other A022553(5) = 25 are primitive.
A000048(5) = 3 among those are self-complementary:
 0000011111
 0001011101
 0010011011
The remaining 22 necklaces form a(5) = 11 complement pairs:
 0000101111 0000111101
 0000110111 0001111001
 0000111011 0001001111
 0001010111 0001110101
 0001011011 0010011101
 0001100111 0001110011
 0001101011 0010100111
 0001101101 0010010111
 0010101011 0011010101
 0010101101 0010110101
 0010110011 0011001101

Tilman Piesk, Table of n, a(n) for n = 0..1000

a(n) = (A022553(n) - A000048(n)) / 2.

A386946 a(n) is the number of imprimitive (periodic) 2n-bead balanced binary necklaces.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 5, 1, 10, 4, 27, 1, 88, 1, 247, 29, 810, 1, 2780, 1, 9260, 249, 32067, 1, 113520, 26, 400025, 2704, 1432868, 1, 5179905, 1, 18784170, 32069, 68635479, 271, 252201136, 1, 930138523, 400027, 3446168660, 1, 12817096533, 1, 47820447036, 5173304

Offset: 0

Tilman Piesk, Aug 10 2025

A003239(n) is the number of 2n-bead balanced binary necklaces. A022553(n) among them are primitive.
The remaining a(n) necklaces are periodic.
Sequences counting 2n-bead balanced binary necklaces:
                       primitive  imprimitive
                     +-----------------------+---------+
  self-complementary |  A000048     A115118  | A000013 |
   complement pairs  |  A383904     A387130  | A386388 |
                     +-----------------------+---------+
                     |  A022553      this    | A003239 |
                     +-----------------------+---------+

			  n | A003239(n) A022553(n) | a(n)
  0 |         1          1  |   0
  1 |         1          1  |   0
  2 |         2          1  |   1
  3 |         4          3  |   1
  4 |        10          8  |   2
  5 |        26         25  |   1
  6 |        80         75  |   5
  7 |       246        245  |   1
  8 |       810        800  |  10
  9 |      2704       2700  |   4
 10 |      9252       9225  |  27
 11 |     32066      32065  |   1
 12 |    112720     112632  |  88
 13 |    400024     400023  |   1
 14 |   1432860    1432613  | 247
 15 |   5170604    5170575  |  29
 16 |  18784170   18783360  | 810
There are A003239(8) = 810 balanced binary necklaces of length 16. A022553(8) = 800 of them are primitive. a(n) = 10 are not. See A387130 for a list.

Tilman Piesk, Table of n, a(n) for n = 0..1000

a(n) = A003239(n) - A022553(n).

a(n) = A115118(n) + 2 * A387130(n).

A387130 a(n) is the number of complement pairs of imprimitive (periodic) 2n-bead balanced binary necklaces.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 3, 1, 11, 0, 39, 0, 118, 12, 395, 0, 1372, 0, 4601, 119, 15986, 0, 56662, 11, 199854, 1337, 716135, 0, 2589376, 0, 9391051, 15987, 34315811, 129, 126096824, 0, 465062362, 199855, 1723071186, 0, 6408523001, 0, 23910175807, 2586090, 89493721076

Offset: 0

Tilman Piesk, Aug 17 2025

A386946(n) is the number of primitive 2n-bead balanced binary necklaces (corresponding to Lyndon words), and A115118 is the number of those that are self-complementary (i.e., can be rotated so that all beads change color). Their difference 2*a(n) is the number of those that are not self-complementary. a(n) is the number pairs of distinct complements.
Doubled entries: 0, 0, 0, 0, 0, 0, 2, 0, 6, 2, 22, 0, 78, 0, 236, 24, 790, 0, 2744, ...
Sequences counting 2n-bead balanced binary necklaces:
                       primitive  imprimitive
                     +-----------------------+---------+
  self-complementary |  A000048     A115118  | A000013 |
   complement pairs  |  A383904      this    | A386388 |
                     +-----------------------+---------+
                     |  A022553     A386946  | A003239 |
                     +-----------------------+---------+

			  n | A386946(n) A115118(n) | 2*a(n)    a(n) | A386388(n) A383904(n)
  0 |         0          0  |     0       0  |         0          0
  1 |         0          0  |     0       0  |         0          0
  2 |         1          1  |     0       0  |         0          0
  3 |         1          1  |     0       0  |         1          1
  4 |         2          2  |     0       0  |         3          3
  5 |         1          1  |     0       0  |        11         11
  6 |         5          3  |     2       1  |        36         35
  7 |         1          1  |     0       0  |       118        118
  8 |        10          4  |     6       3  |       395        392
  9 |         4          2  |     2       1  |      1337       1336
 10 |        27          5  |    22      11  |      4598       4587
 11 |         1          1  |     0       0  |     15986      15986
 12 |        88         10  |    78      39  |     56270      56231
 13 |         1          1  |     0       0  |    199854     199854
 14 |       247         11  |   236     118  |    716132     716014
 15 |        29          5  |    24      12  |   2584754    2584742
 16 |       810         20  |   790     395  |   9391051    9390656
Examples for n=8 with necklaces of length 16:
The total number of necklaces is A003239(8) = 810.
A022553(8) = 800 of them are primitive.
The other A386946(8) = 10 are periodic.
A115118(8) = 4 among those are self-complementary:
 0000111100001111
 0010110100101101
 0011001100110011
 0101010101010101
The remaining 6 necklaces form a(8) = 3 complement pairs:
 0001011100010111 0001110100011101
 0001101100011011 0010011100100111
 0010101100101011 0011010100110101

Tilman Piesk, Table of n, a(n) for n = 0..1000

Cf. A386946, A115118, A386388, A383904.

a(n) = (A386946(n) - A115118(n)) / 2.

a(n) = A386388(n) - A383904(n).

cp's OEIS Frontend

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.

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A383904 a(n) is the number of complement pairs of primitive 2n-bead balanced binary necklaces.

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A386946 a(n) is the number of imprimitive (periodic) 2n-bead balanced binary necklaces.

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A387130 a(n) is the number of complement pairs of imprimitive (periodic) 2n-bead balanced binary necklaces.

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