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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.
0
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
1,4
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.
EXAMPLE
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
KEYWORD
nonn,base,tabl
AUTHOR
Peter Kagey, Oct 04 2024
STATUS
approved