A376789 Table read by antidiagonals: T(n,k) is the number of Lyndon words of length k on the alphabet {0,1} whose prefix is the bitwise complement of the binary expansion of n with n >= 1 and k >= 1.

1, 1, 0, 2, 1, 0, 3, 1, 0, 0, 6, 1, 1, 0, 0, 9, 2, 2, 1, 0, 0, 18, 2, 4, 1, 0, 0, 0, 30, 4, 7, 1, 0, 1, 0, 0, 56, 5, 14, 1, 1, 1, 0, 0, 0, 99, 8, 25, 2, 1, 2, 1, 0, 0, 0, 186, 11, 48, 2, 2, 3, 2, 1, 0, 0, 0, 335, 18, 88, 3, 3, 6, 4, 1, 0, 0, 0, 0

Offset: 1

Author

Peter Kagey, Oct 04 2024

Keywords

• nonn
• base
• tabl

Comments

T(n,k) = 0 if n is in A366195.

Row 1 is A059966.

Row 2 is A006206 for n > 1.

Row 3 is A065491 for n > 2.

Row 4 is A065417.

Row 6 is A349904.

Examples

			Table begins
n\k| 1  2  3  4  5  6   7   8   9  10   11   12
---+-------------------------------------------
 1 | 1, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335
 2 | 0, 1, 1, 1, 2, 2,  4,  5,  8, 11,  18,  25
 3 | 0, 0, 1, 2, 4, 7, 14, 25, 48, 88, 168, 310
 4 | 0, 0, 1, 1, 1, 1,  2,  2,  3,  4,   6,   7
 5 | 0, 0, 0, 0, 1, 1,  2,  3,  5,  7,  12,  18
 6 | 0, 0, 1, 1, 2, 3,  6, 10, 18, 31,  56,  96
 7 | 0, 0, 0, 1, 2, 4,  8, 15, 30, 57, 112, 214
 8 | 0, 0, 0, 1, 1, 1,  1,  1,  2,  2,   3,   3
 9 | 0, 0, 0, 0, 0, 0,  1,  1,  1,  2,   3,   4
10 | 0, 0, 0, 0, 1, 1,  2,  3,  5,  7,  12,  18
11 | 0, 0, 0, 0, 0, 0,  0,  0,  0,  0,   0,   0
12 | 0, 0, 0, 1, 1, 2,  3,  5,  9, 15,  26,  43
T(6,5) = 2 because 6 is 110 in base 2, its bitwise complement is 001, and there are T(6,5) = 2 length-5 Lyndon words that begin with 001: 00101 and 00111.
		

Crossrefs

Cf. A059966, A006206, A065491, A065417, A349904.

Cf. A365746.