A323908 Reversing binary representation of A004718, Per Nørgård's "infinity sequence".

0, 1, 3, 2, 1, 0, 6, 7, 3, 2, 0, 1, 2, 3, 5, 4, 1, 0, 6, 7, 0, 1, 3, 2, 6, 7, 1, 0, 7, 6, 12, 13, 3, 2, 0, 1, 2, 3, 5, 4, 0, 1, 3, 2, 1, 0, 6, 7, 2, 3, 5, 4, 3, 2, 0, 1, 5, 4, 2, 3, 4, 5, 15, 14, 1, 0, 6, 7, 0, 1, 3, 2, 6, 7, 1, 0, 7, 6, 12, 13, 0, 1, 3, 2, 1, 0, 6, 7, 3, 2, 0, 1, 2, 3, 5, 4, 6, 7, 1, 0, 7, 6, 12, 13, 1, 0

Offset: 0

Author

Antti Karttunen, Feb 09 2019

Keywords

• nonn

Comments

The composer Per Nørgård's name is also written in the OEIS as Per Noergaard.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65536
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A004718, A048724, A065620, A065621, A083866 (positions of zeros), A323907 (rgs-transform), A323909.

Programs

PARI

up_to = 65536;
A004718list(up_to) = { my(v=vector(up_to)); v[1]=1; v[2]=-1; for(n=3, up_to, v[n] = if(n%2, 1+v[n>>1], -v[n/2])); (v); }; \\ After code in A004718.
v004718 = A004718list(up_to);
A004718(n) = if(!n,n,v004718[n]);
A048724(n) = bitxor(n, n<<1);
A065621(n) = bitxor(n-1,n+n-1);
A323908(n) = if(A004718(n)<=0, A048724(-A004718(n)), A065621(A004718(n)));

Python

from itertools import groupby
def A323908(n):
    c = 0
    for k, g in groupby(bin(n)[2:]):
        c = c+len(list(g)) if k == '1' else (-c if len(list(g))&1 else c)
    return -c^(-c<<1) if c<=0 else c^(c&~-c)<<1 # Chai Wah Wu, Mar 02 2023

Formula

If A004718(n) <= 0, a(n) = A048724(-A004718(n)), otherwise a(n) = A065621(A004718(n)).

For all n >= 1, A065620(a(n)) = A004718(n).