A074261 -id:A074261 - cp's OEIS frontend

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

Clark Kimberling

nonn

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

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

Cf. A000002, A074261, A022293, A342799.

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

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 *)

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

0 prepended by Jon Perry, Sep 11 2012

A342799 Numbers m such that there are more 1s than 2s in {K(1), .., K(m)}, where K = A000002 (Kolakoski sequence).

Original entry on oeis.org

1, 5, 7, 17, 23, 25, 29, 31, 32, 33, 34, 35, 37, 41, 43, 49, 51, 52, 53, 55, 59, 61, 71, 73, 79, 125, 131, 133, 134, 135, 137, 141, 143, 149, 161, 179, 185, 187, 191, 193, 194, 195, 196, 197, 199, 201, 202, 203, 205, 209, 211, 221, 223, 227, 229, 230, 231

Offset: 1

Clark Kimberling, May 10 2021

nonn

The three sequences A022292 (without the initial 0), A074261, and A342799 partition the positive integers.

			The Kolakoski sequence is (1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, ...), so that a(1) = 1, a(2) = 5.

Cf. A000002, A022292, A022293, A074261.

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 *)

cp's OEIS Frontend

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

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