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).
From n = 128 on, a(n) > A010097(n), but the terms of the two sequences remain approximately the same size up to n = 511. - M. F. Hasler, Oct 24 2024
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"
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
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.
PROG
(PARI) apply( {A171885(n)=if(n>1, n-3<<(n=exponent(n))+2*4^n, n)}, [0..44]) \\ M. F. Hasler, Oct 24 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Robert Munafo, Mar 19 2010
STATUS
approved