A022292 Exactly half of first a(n) terms of Kolakoski sequence A000002 are 1's (not known to be infinite).

0, 2, 4, 6, 8, 10, 14, 16, 18, 20, 22, 24, 26, 28, 30, 36, 38, 40, 42, 44, 46, 48, 50, 54, 56, 58, 60, 62, 64, 68, 70, 72, 74, 76, 78, 80, 82, 86, 88, 98, 104, 106, 116, 118, 122, 124, 126, 128, 130, 132, 136, 138, 140, 142, 144, 146, 148, 150, 152, 158

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

The sequences A022292, A074261, and A342799 partition the nonnegative integers. - Clark Kimberling, May 10 2021

Links

• Joerg Arndt, Table of n, a(n) for n = 0..8739

Crossrefs

Cf. A000002, A074261, A022293, A342799.

Programs

JavaScript

a=new Array();
a[1]=1; a[2]=2; a[3]=2; cd=1; ap=3;
for (i=4; i<1000; i++)
{
    if (a[ap]==1) a[i]=cd;
    else {a[i]=cd; a[i+1]=cd; i++}
    ap++;
    cd=3-cd;
}
oc=0; tc=0;
for (i=1; i<1000; i++)
{
    if (oc==tc) document.write(i-1+", ");
    if (a[i]==1) oc++;
    else tc++;
}
// Jon Perry, Sep 11 2012

Mathematica

k = Prepend[Nest[Flatten[Partition[#, 2] /. {{2, 2} -> {2, 2, 1, 1}, {2, 1} -> {2, 2, 1}, {1, 2} -> {2, 1, 1}, {1, 1} -> {2, 1}}] &, {2, 2}, 14], 1]; (* A000002 *)
Select[Range[400], Count[Take[k, #], 1] < #/2 &]   (* A074261 *)
Select[Range[400], Count[Take[k, #], 1] == #/2 &]  (* A022292 *)
Select[Range[400], Count[Take[k, #], 1] > #/2 &]   (* A342799 *)
(* Clark Kimberling, May 10 2021 *)

Formula

Conjecture: a(n) is asymptotic to c*n*log(n) for some constant c <= 1. - Benoit Cloitre, Nov 17 2003

Extensions

0 prepended by Jon Perry, Sep 11 2012