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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.
(history; published version)

#11 by Peter Kagey at Sun Oct 13 17:59:47 EDT 2024

STATUS

editing

proposed

Discussion

Wed Oct 16

21:36

N. J. A. Sloane: [Annoucement: The Sequence Fans Mailing List is now a Google Group: Sign into Google, go to https://groups.google.com/g/seqfan, click "Join this group" - Neil Sloane]

#10 by Peter Kagey at Sun Oct 06 13:20:19 EDT 2024

NAME

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.

EXAMPLE

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.

KEYWORD

nonn,changed,base,tabl

Discussion

Sun Oct 13

17:56

OEIS Server: This sequence has not been edited or commented on for a week
yet is not proposed for review.  If it is ready for review, please
visit https://oeis.org/draft/A376789 and click the button that reads
"These changes are ready for review by an OEIS Editor."

Thanks.
  - The OEIS Server

#9 by Peter Kagey at Sun Oct 06 13:18:29 EDT 2024

EXAMPLE

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, ...

#8 by Peter Kagey at Sun Oct 06 12:50:57 EDT 2024

CROSSREFS

Cf. A059966 (row 1), , A006206 (row 2), , A065491 (row 3), , A065417 (row 4), , A349904 (row 6),.

#7 by Peter Kagey at Sun Oct 06 12:50:24 EDT 2024

CROSSREFS

Cf. A059966 (row 1)., A006206 (row 2), A065491 (row 3), A065417 (row 4), A349904 (row 6),

#6 by Peter Kagey at Sun Oct 06 12:47:49 EDT 2024

CROSSREFS

Cf. A059966 (row 1).

Cf. A365746.

#5 by Peter Kagey at Fri Oct 04 01:25:17 EDT 2024

COMMENTS

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

#4 by Peter Kagey at Fri Oct 04 01:20:57 EDT 2024

COMMENTS

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.

CROSSREFS

A059966 (row 1)

#3 by Peter Kagey at Fri Oct 04 01:14:08 EDT 2024

NAME

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

#2 by Peter Kagey at Fri Oct 04 01:13:21 EDT 2024

NAME

allocated for Peter Kagey

Table read by antidiagonals: T(n,k) is the number of Lyndon words of length k whose prefix is the bitwise complement of the binary expansion of n.

DATA

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

EXAMPLE

Table begins

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, ...

KEYWORD

allocated

nonn

AUTHOR

Peter Kagey, Oct 04 2024

STATUS

approved

editing