A361477 a(n) is the number of integers whose binary expansions have the same multiset of run-lengths as that of n.

1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 3, 1, 3, 2, 1, 2, 3, 4, 3, 4, 1, 4, 3, 2, 3, 4, 3, 2, 3, 2, 1, 2, 3, 4, 6, 6, 5, 6, 6, 4, 5, 1, 5, 6, 5, 4, 3, 2, 6, 6, 1, 6, 5, 6, 6, 1, 6, 4, 6, 2, 3, 2, 1, 2, 3, 4, 6, 12, 5, 12, 3, 12, 10, 6, 10, 4, 10, 12, 6, 4, 5, 6, 10

Offset: 0

Author

Rémy Sigrist, Mar 13 2023

Keywords

• nonn
• base

Comments

This sequence has similarities with A090706; here we consider multisets of run-lengths, there multisets of digits in binary expansions.

Examples

			For n = 18:
- the binary expansion of 18 is "10010",
- the corresponding multiset of run-lengths is m = (1, 2, 1, 1),
- m has 4 terms: 3 times "1" and once "2",
- so a(18) = 4! / (3! * 1!) = 4.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8192
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A090706, A101211, A140690.

Programs

PARI

a(n) = { my (r=[]); while (n, my (v=valuation(n+n%2, 2)); n\=2^v; r=concat(v, r)); my (s=Set(r), f=vector(#s)); for (k=1, #r, f[setsearch(s, r[k])]++); (#r)! / prod(k=1, #f, f[k]!) }

Python

from math import factorial, prod
from itertools import groupby
from collections import Counter
def A361477(n): return factorial(len(c:=[len(list(g)) for k, g in groupby(bin(n)[2:])]))//prod(map(factorial,Counter(c).values())) # Chai Wah Wu, Mar 16 2023

Formula

a(n) = 1 iff n = 0 or n belongs to A140690.