A140690 -id:A140690 - cp's OEIS frontend

A336963 Left-rotate run-lengths of consecutive equal digits in binary representation of n.

0, 1, 2, 3, 6, 5, 4, 7, 14, 13, 10, 9, 12, 11, 8, 15, 30, 29, 26, 25, 22, 21, 18, 17, 28, 27, 20, 19, 24, 23, 16, 31, 62, 61, 58, 57, 54, 53, 50, 49, 46, 45, 42, 41, 38, 37, 34, 33, 60, 59, 52, 51, 44, 43, 36, 35, 56, 55, 40, 39, 48, 47, 32, 63, 126, 125, 122

Offset: 0

Rémy Sigrist, Aug 09 2020

This sequence is a permutation of the nonnegative integers, with inverse A336962.

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     2      10         10
   3     3      11         11
   4     6     100        110
   5     5     101        101
   6     4     110        100
   7     7     111        111
   8    14    1000       1110
   9    13    1001       1101
  10    10    1010       1010
  11     9    1011       1001
  12    12    1100       1100
  13    11    1101       1011
  14     8    1110       1000
  15    15    1111       1111

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Rémy Sigrist, Colored scatterplot of the first 2^16 terms (where the color is function of A090996(n))
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A006257, A090996, A101211, A140690, A336962.

toruns(n) = { my (r=[]); while (n, my (v=valuation(n+n%2,2)); n\=2^v; r=concat(v,r)); r }
fromruns(r) = { my (v=0); for (k=1, #r, v=(v+k%2)*2^r[k]-k%2); v }
a(n) = { my (r=toruns(n)); fromruns(vector(#r, k, r[1+k%#r])) }

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

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

Original entry on oeis.org

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

Rémy Sigrist, Mar 13 2023

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

			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.

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

Cf. A090706, A101211, A140690.

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]!) }

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

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

cp's OEIS Frontend

A336963 Left-rotate run-lengths of consecutive equal digits in binary representation of n.

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