A335599 Sequence is limit_{k->oo} s_k, where s_k = s_{k-1}, s_{k-1}[k-1] + 2^(k-1), ..., s_{k-1}[end] + 2^(k-1) starting with s_0 = s_0[0..1] = 0,0.

0, 0, 1, 1, 2, 3, 3, 5, 5, 6, 7, 7, 9, 10, 11, 11, 13, 13, 14, 15, 15, 18, 19, 19, 21, 21, 22, 23, 23, 25, 26, 27, 27, 29, 29, 30, 31, 31, 35, 35, 37, 37, 38, 39, 39, 41, 42, 43, 43, 45, 45, 46, 47, 47, 50, 51, 51, 53, 53, 54, 55, 55, 57, 58, 59, 59, 61, 61

Offset

0,5

Comments

When written in binary, the terms are 0, 0, 1, 1, 10, 11, 11, 101, 101, 110, 111, 111, 1001, 1010, 1011, 1101, 1110, 1111, 1111, 10010, 10011, 10011, 10101, ...

a(n) = m is the smallest solution to m + bitcount(m) = n or n-1. So a(n) = smaller nonzero of A228086(n) and A228086(n-1) (for n>=2). - Kevin Ryde, Jul 05 2020

Links

Table of n, a(n) for n=0..67.

Kevin Ryde, PARI/GP code and explanation, quantity "b(n)".

Formula

a(n) + bitcount(a(n)) + A334820(n) = n for n>=0.

Maple

s:= proc(n) option remember; `if`(n=0, [0, 0][], (l->
     [l[], map(x-> x+2^(n-1), l[n..-1])[]][])([s(n-1)]))
    end:
s(7);  # gives 136 = A005126(7) terms;  # Alois P. Heinz, Jul 04 2020

Mathematica

a[n_] := If[n == 0, 0,
Module[{m = n, k = Floor@Log2[n]}, m -= k + 1; While[k >= 0,
     If[BitGet[m, k] == 0, m++;
     If[BitGet[m, k] == 1, Return[m-1]]]; k--]; m]];
Table[a[n], {n, 0, 67}] (* Jean-François Alcover, May 30 2022, after Kevin Ryde *)

Prog

(PARI) a(n) = { if(n, my(k=logint(n, 2)); n-=k+1;
  while(k>=0, if(!bittest(n, k), n++; if(bittest(n, k), return(n-1))); k--));
  n; }  \\ Kevin Ryde, Jul 05 2020

Crossrefs

Cf. A005126, A334820.

Sequence in context: A023816 A353714 A159237 * A227065 A010761 A320840

Adjacent sequences: A335596 A335597 A335598 * A335600 A335601 A335602

Keyword

nonn

Author

Richard Aime Blavy, Jun 15 2020

Status

approved