|
|
A281150
|
|
Elias delta code for n.
|
|
6
|
|
|
1, 1000, 1001, 10100, 10101, 10110, 10111, 11000000, 11000001, 11000010, 11000011, 11000100, 11000101, 11000110, 11000111, 110010000, 110010001, 110010010, 110010011, 110010100, 110010101, 110010110, 110010111, 110011000, 110011001, 110011010
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
The number of bits in a(n) is equal to A140341(n).
a(n) is the prefix-free encoding of n-1 defined on pages 180-181 of Shallit (2008). - N. J. A. Sloane, Mar 18 2019
|
|
REFERENCES
|
Shallit, Jeffrey. A second course in formal languages and automata theory. Cambridge University Press, 2008. See E(m) on page 181. - N. J. A. Sloane, Mar 18 2019
|
|
LINKS
|
|
|
FORMULA
|
For a given integer n, a(n) is composed of two parts. The first part equals 1+floor(log_2 n) and the second part equals n-2^(floor(log_2 n)). The first part is stored in Elias Gamma Code and the second part is stored in a binary using floor(log_2 n) bits. The first and the second parts are concatenated to give a(n).
|
|
EXAMPLE
|
For n = 9, the first part is "11000" and the second part is "001". So a(9) = 11000001.
|
|
PROG
|
(Python)
def unary(n):
....return "1"*(n-1)+"0"
def elias_gamma(n):
....if n==1:
........return "1"
....k=int(math.log(n, 2))
....fp=unary(1+k) #fp is the first part
....sp=n-2**(k) #sp is the second part
....nb=k #nb is the number of bits used to store sp in binary
....sp=bin(sp)[2:]
....if len(sp)<nb:
........sp=("0"*(nb-len(sp)))+sp
....return fp+sp
def elias_delta(n):
....if n==1:
........return "1"
....k=int(math.log(n, 2))
....fp=elias_gamma(1+k)#fp is the first part
....sp=n-2**(k) #sp is the second part
....nb=k #nb is the number of bits used to store sp in binary
....sp=bin(sp)[2:]
....if len(sp)<nb:
........sp=("0"*(nb-len(sp)))+sp
....return fp+sp
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,base
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|