A056791 Weight of binary expansion of n + length of binary expansion of n.
1, 2, 3, 4, 4, 5, 5, 6, 5, 6, 6, 7, 6, 7, 7, 8, 6, 7, 7, 8, 7, 8, 8, 9, 7, 8, 8, 9, 8, 9, 9, 10, 7, 8, 8, 9, 8, 9, 9, 10, 8, 9, 9, 10, 9, 10, 10, 11, 8, 9, 9, 10, 9, 10, 10, 11, 9, 10, 10, 11, 10, 11, 11, 12, 8, 9, 9, 10, 9, 10, 10, 11, 9, 10, 10, 11, 10, 11, 11, 12, 9, 10, 10, 11, 10, 11, 11
Offset: 0
Examples
12 = 1100 in binary, so a(12)=2+4=6.
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..16384
- Ralf Stephan, Some divide-and-conquer sequences ...
- Ralf Stephan, Table of generating functions
- Index to sequences related to the complexity of n
Programs
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Mathematica
Table[If[n==0,1,s=IntegerDigits[n,2];Total@s+Length@s],{n,0,100}] (* Giorgos Kalogeropoulos, Sep 13 2021 *) -
PARI
a(n) = if (n==0, 1, my(b=binary(n)); vecsum(b) + #b); \\ Michel Marcus, Sep 13 2021
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Python
def a(n): b = bin(n)[2:]; return b.count('1') + len(b) print([a(n) for n in range(87)]) # Michael S. Branicky, Sep 13 2021
Formula
a(n) = a((n - n mod 2) / (2 - n mod 2)) + 1 for n>0, a(0)=1. - Reinhard Zumkeller, Jul 29 2002
a(2n) = a(n)+1, a(2n+1) = a(n)+2. G.f.: 1 + 1/(1-x) * sum(k>=0, (2t+t^2)/(1+t), t=x^2^k). For n>0, a(n) = 2*A000120(n) + A080791(n) = A000120(n) + A029837(n). - Ralf Stephan, Jun 14 2003
Extensions
More terms from James Sellers, Sep 06 2000 and from David W. Wilson, Sep 07 2000