A080679 -id:A080679 - cp's OEIS frontend

A169671 Lexicographically earliest de Bruijn sequence for n = 6 and k = 2.

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0

Offset: 0

N. J. A. Sloane, Apr 11 2010

			Periodic with period 64, the period being:
0000001000011000101000111001001011001101001111010101110110111111.

Ray Chandler, Table of n, a(n) for n = 0..1023
Frank Ruskey, Generate de Bruijn sequences
Index entries for linear recurrences with constant coefficients, signature (1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1).

Cf. A021913, A169675, A080679, A169672, A169671, A169673, A169674.

See A058342 for another version.

A169673 Lexicographically earliest de Bruijn sequence for n = 7 and k = 2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1

Offset: 0

N. J. A. Sloane, Apr 11 2010

			Periodic with period 128, the period being:
00000001000001100001010000111000100100010110001101000111100100110\
010101001011100110110011101001111101010110101111011011101111111

Ray Chandler, Table of n, a(n) for n = 0..1023
Frank Ruskey, Generate de Bruijn sequences
Index entries for linear recurrences with constant coefficients, order 128.

Cf. A021913, A169675, A080679, A169672, A169671, A169674.

A169674 Lexicographically earliest de Bruijn sequence for n = 8 and k = 2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0

Offset: 0

N. J. A. Sloane, Apr 11 2010

			Periodic with period 256, the period being:
0000000010000001100000101000001110000100100001011000011010000111100010\
0010011000101010001011100011001000110110001110100011111001001010010011\
1001010110010110100101111001100110101001101110011101100111101001111110\
1010101110101101101011111011011110111011111111

Ray Chandler, Table of n, a(n) for n = 0..1023
Frank Ruskey, Generate de Bruijn sequences
Index entries for linear recurrences with constant coefficients, order 256.

Cf. A021913, A169675, A080679, A169672, A169671, A169673.

A169672 Lexicographically earliest de Bruijn sequence for n = 5 and k = 2.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0

Offset: 0

N. J. A. Sloane, Apr 11 2010

			Periodic with period 32, the period being: 00000100011001010011101011011111.

Frank Ruskey, Generate de Bruijn sequences
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A021913, A169675, A080679, A169671, A169673, A169674.

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},{0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1},99] (* Ray Chandler, Aug 26 2015 *)

A169675 Lexicographically earliest de Bruijn sequence for n = 3 and k = 2.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0

Offset: 0

N. J. A. Sloane, Apr 11 2010

The lexicographically earliest de Bruijn sequence for n = 2 and k = 2 is 0011 repeated (see A021913).

			Periodic with period 8, the period being 00010111.

Frank Ruskey, Generate de Bruijn sequences
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Cf. A021913, A080679, A169672, A169671, A169673, A169674.

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1},{0, 0, 0, 1, 0, 1, 1, 1},99] (* Ray Chandler, Aug 25 2015 *)
PadRight[{},120,{0,0,0,1,0,1,1,1}] (* Harvey P. Dale, Aug 01 2024 *)

A169676 Lexicographically earliest de Bruijn sequence for n = 2 and k = 3.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 1, 2, 2, 0, 0, 1, 0, 2, 1, 1, 2, 2, 0, 0, 1, 0, 2, 1, 1, 2, 2, 0, 0, 1, 0, 2, 1, 1, 2, 2, 0, 0, 1, 0, 2, 1, 1, 2, 2, 0, 0, 1, 0, 2, 1, 1, 2, 2, 0, 0, 1, 0, 2, 1, 1, 2, 2, 0, 0, 1, 0, 2, 1, 1, 2, 2, 0, 0, 1, 0, 2, 1, 1, 2, 2, 0, 0, 1, 0, 2, 1, 1, 2, 2, 0, 0, 1, 0, 2, 1, 1, 2, 2

Offset: 0

N. J. A. Sloane, Apr 11 2010

			Periodic with period 9, the period being 001021122.

Frank Ruskey, Generate de Bruijn sequences
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Cf. A021913, A169675, A080679, A169672, A169671, A169673, A169674.

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 1},{0, 0, 1, 0, 2, 1, 1, 2, 2},99] (* Ray Chandler, Aug 26 2015 *)

If someone would like to help, I would like to get analogous entries for k = 3 and n = 3,4,5,6; k = 4 and n = 2,3,4,5,6; k = 5 and n = 2,3,4,5,6; and n = 2 and k = 6,7,8,9, ...

A308831 Start with generation 0, which is the empty sequence. For generation N>=1, extend the existing sequence into a non-cyclic ternary de Bruijn sequence of order N. If more than one extension is possible, choose the lexicographically earliest.

Original entry on oeis.org

0, 1, 2, 0, 0, 2, 2, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 2, 1, 0, 2, 0, 2, 1, 2, 2, 2, 0, 1, 0, 0, 1, 1, 0, 1, 0, 2, 1, 0, 0, 2, 0, 0, 0, 0, 2, 1, 1, 1, 1, 0, 2, 2, 0, 0, 1, 2, 1, 1, 2, 0, 1, 1, 2, 2, 0, 2, 0, 1, 2, 2, 1, 2, 1, 2, 0, 2, 2, 2, 2, 1, 0, 1, 2

Offset: 0

A. D. Skovgaard, Jun 27 2019

nonn

If using a binary alphabet instead, it would not be possible to extend the sequence infinitely as a de Bruijn sequence (order 3 needs an extra term: 01100010111). - A. D. Skovgaard, Apr 19 2020

			Generation 1:
[012] (All ternary sequences of length 1 now appear. With 3! = 6 solutions, the lexicographically earliest is chosen.)
Generation 2:
[0120022110] (The sequence is extended from the previous generation, now including all ternary sequences of length 2.)
The process continues.

A. D. Skovgaard, Table of n, a(n) for n = 0..246
A. D. Skovgaard, Python program to generate the sequence, with explanatory comments
A. D. Skovgaard, a(n) for n = 0..246 in compressed notation

Cf. A080679 (binary equivalent), A166315, A169676.

A144569 A de Bruijn sequence B(4,3) found by Ted Bell.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 2, 0, 2, 2, 2, 1, 2, 1, 1, 2, 2, 0, 0, 3, 0, 3, 3, 3, 2, 3, 2, 2, 3, 3, 1, 3, 1, 1, 3, 3, 0, 1, 3, 2, 0, 3, 2, 1, 0, 3, 1, 0, 2, 3, 1, 2, 0, 1, 2, 3, 0, 2, 1, 3, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 2, 0, 2, 2, 2, 1, 2, 1, 1, 2, 2, 0, 0, 3, 0, 3, 3, 3, 2, 3, 2, 2, 3, 3, 1

Offset: 0

N. J. A. Sloane, Apr 09 2010, based on a communication from David Paterson (David.Paterson(AT)csiro.au)

			Period 64: 0001110100202221211220030333232233131133013203210310231201230213

Mark Dow, Minimal arrays containing all sub-array combinations of symbols: de Bruijn sequences and tori (Web archive)
Wikipedia, De Bruijn Sequences
Index entries for linear recurrences with constant coefficients, order 64.

A135472 gives an example of a B(9,2).

Cf. also A080679, A058342, A083570, A166315, A166316.

cp's OEIS Frontend

A169671 Lexicographically earliest de Bruijn sequence for n = 6 and k = 2.

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A169673 Lexicographically earliest de Bruijn sequence for n = 7 and k = 2.

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A169674 Lexicographically earliest de Bruijn sequence for n = 8 and k = 2.

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A169672 Lexicographically earliest de Bruijn sequence for n = 5 and k = 2.

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A169675 Lexicographically earliest de Bruijn sequence for n = 3 and k = 2.

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A169676 Lexicographically earliest de Bruijn sequence for n = 2 and k = 3.

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Extensions

A308831 Start with generation 0, which is the empty sequence. For generation N>=1, extend the existing sequence into a non-cyclic ternary de Bruijn sequence of order N. If more than one extension is possible, choose the lexicographically earliest.

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A144569 A de Bruijn sequence B(4,3) found by Ted Bell.

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