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
Keywords
Examples
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.
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"
Crossrefs
Programs
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PARI
apply( {A171885(n)=if(n>1, n-3<<(n=exponent(n))+2*4^n, n)}, [0..44]) \\ M. F. Hasler, Oct 24 2024
Formula
a(n) = n + 2*4^L - 3*2^L with L = floor(log_2(n)), for n > 1. - M. F. Hasler, Oct 24 2024
Comments