|
|
A129760
|
|
Bitwise AND of binary representation of n-1 and n.
|
|
29
|
|
|
0, 0, 2, 0, 4, 4, 6, 0, 8, 8, 10, 8, 12, 12, 14, 0, 16, 16, 18, 16, 20, 20, 22, 16, 24, 24, 26, 24, 28, 28, 30, 0, 32, 32, 34, 32, 36, 36, 38, 32, 40, 40, 42, 40, 44, 44, 46, 32, 48, 48, 50, 48, 52, 52, 54, 48, 56, 56, 58, 56, 60, 60, 62, 0, 64, 64, 66, 64, 68, 68, 70, 64, 72, 72, 74
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
Also the number of Ducci sequences with period n.
a(n) is the parent of vertex n in the binomial tree. The binomial tree is root vertex n=0, then for n>=1 the parent of n is n with its least significant 1-bit changed to a 0-bit. Binomial tree order 5, n=0 to 31 inclusive, is the frontispiece of Knuth volume 1, second and subsequent editions. Vertices are shown there with n in binary dots and a(n) is the next vertex towards the root at the bottom of the page. - Kevin Ryde, Jul 24 2019
|
|
REFERENCES
|
Donald E. Knuth, The Art of Computer Programming, volume 1, second edition, frontispiece. Reproduced with brief description of the art in Donald E. Knuth, Selected Papers on Fun and Games, 2010, Chapter 47 Geek Art, figure 16, page 679.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = n AND n-1.
a((2*n-1)*2^p) = (2*n-2)*(2^p), p>=0.
a(2*n-1) = (2*n-2), n>=1, and a(2^p+1) = 2^p, p>=1. (End)
|
|
EXAMPLE
|
a(6) = 6 AND 5 = binary 110 AND 101 = binary 100 = 4.
|
|
MAPLE
|
nmax := 75: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := (2*n-2) * 2^p od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Jun 22 2011, revised Jan 25 2013
|
|
MATHEMATICA
|
|
|
PROG
|
(C) int a(int n) { return n & (n-1); }
(Python)
def a(n): return n & (n-1)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|