OFFSET
0,3
COMMENTS
A simple scatter plot reveals a self-similar structure that resembles flying geese.
Ignoring the initial zero term, split the sequence into rows of increasing binary magnitude such that the terms in row m satisfy 2^m <= a(n) < 2^(m+1).
0: 1,
1: 3,
2: 6,6,7,
3: 13,12,12,13,15,
4: 26,26,27,25,24,24,25,27,30,30,31,
5: 53,52,52,53,55,50,50,51,49,48,48,49,51,54,54,55,61,60,60,61,63,
Then,
Row m starts at n = A005578(m+1) in the original sequence
The first term in row m is A081254(m)
The last term in row m is 2^(m+1)-1
The number of terms in row m is A001045(m+1)
The number of distinct terms in row m is A005578(m)
The number of ascending runs in row m is A005578(m)
The number of non-ascending runs in row m is A005578(m)
The number of descending runs in row m is A052950(m)
The number of non-descending runs in row m is A005578(m-1)
The sum of terms in row m is A178747(m)
The total number of '1' bits in the terms of row n is A178748(m)
LINKS
D. Scambler, Table of n, a(n) for n = 0..1024
FORMULA
If n is a power of 2, a(n) = n*3/2. Lim(a(n)/n) = 3/2.
EXAMPLE
0 -> low bit toggles -> 1 -> should be 2 but low bit does not toggle -> 3 -> should be 4 but 2nd-lowest bit does not toggle -> 6 -> should be 7 but low bit does not toggle -> 6 -> low bit toggles -> 7
PROG
(PARI) seq(n)={my(a=vector(n+1), f=0, p=0); for(i=2, #a, my(b=bitxor(p+1, p)); f=bitxor(f, b); p=bitxor(p, bitand(b, f)); a[i]=p); a} \\ Andrew Howroyd, Mar 03 2020
CROSSREFS
KEYWORD
nonn,look
AUTHOR
David Scambler, Jun 08 2010
STATUS
approved