Richard Aime Blavy - cp's OEIS frontend

User: Richard Aime Blavy

Richard Aime Blavy has authored 2 sequences.

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

Richard Aime Blavy, Jun 15 2020

nonn

In binary 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

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

Cf. A005126, A334820.

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

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 *)

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

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

A334820 Sequence is limit_{t->oo} s_t, where s_k = s_{k-1},s_{k-1}[k-1..end] starting with s_0 = s_0[0..1] = 0,1.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1

Offset: 0

Richard Aime Blavy, May 12 2020

nonn

A nonperiodic sequence of 0 and 1.

Kevin Ryde, PARI/GP code and explanation.

Cf. A005126 (s lengths), A057215.

s:= proc(n) option remember; `if`(n=0, [0, 1][],
      (l-> [l[], l[n..-1][]][])([s(n-1)]))
    end:
s(10);  # gives 1035 = A005126(10) terms;  # Alois P. Heinz, May 12 2020

a[n_] := If[n<2, n, Module[{k = Floor@Log[2, n], m = n}, m-=k+1; While[k >= 0, If[BitGet[m, k]==0, m++; If[BitGet[m, k]==1, Return[1]]]; k--]]; 0];
a /@ Range[0, 104] (* Jean-François Alcover, Nov 16 2020, after Kevin Ryde *)

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(1))); k--)); 0; }  \\ Kevin Ryde, Jun 26 2020

cp's OEIS Frontend

User: Richard Aime Blavy

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.

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