A171885 Representation of n in D. E. Knuth's second prefix-unambiguous, order-preserving binary string system.

0, 1, 4, 5, 24, 25, 26, 27, 112, 113, 114, 115, 116, 117, 118, 119, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 1984, 1985, 1986, 1987, 1988, 1989, 1990, 1991, 1992, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001

Offset

0,3

Comments

The first two terms are the strings "00" and "01"; all others are binary strings beginning with "1".

With the important exception of a(1) = 1, when expressed in binary, any value not appearing in this sequence appears as an initial substring of later terms. For example, 6 (110) is an initial substring of 27 (11011).

References

Donald E. Knuth, "Supernatural Numbers", in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 310-325.

Donald E. Knuth, Selected Papers on Fun and Games. Stanford, California: Center for the Study of Language and Information (2011).

Links

Indranil Ghosh, Table of n, a(n) for n = 0..10000

Robert Munafo, Alternative Number Formats, section on "Lexicographic Strings"

Example

The representations start: 0 = "00"; 1 = "01"; 2 = "100"; 3 = "101"; 4 = "11000"; 5 = "11001"; 6 = "11010"; 7 = "11011"; 8 = "1110000"; 9 = "1110001"; and so on. See the references for longer lists and fuller explanation.

Crossrefs

Knuth's first system gives A000918 (less its initial term).

Knuth's third system is A010097, the Levenshtein codes.

Sequence in context: A010302 A361981 A338422 * A331261 A063986 A039583

Adjacent sequences: A171882 A171883 A171884 * A171886 A171887 A171888

Keyword

nonn

Author

Robert Munafo, Mar 19 2010

Status

approved