A341334 Concatenation of all 01-words, in the order induced by A016777; see Comments.

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

Offset

1

Comments

Let s = (s(n)) be a strictly increasing sequence of positive integers with infinite complement, t = (t(n)).

For n >= 1, let s'(n) be the number of s(i) that are <= n-1 and let t'(n) be the number of t(i) that are <= n-1.

Define w(1) = 0, w(t(1)) = 1, and w(n) = 0w(s'(n)) if n is in s, and w(n) = 1w(t'(n)) if n is in t. Then (w(n)) is the "s-induced ordering" of all 01-words.

s = A016777; t = A007494; s' = A002264; t' = A004523;

In the following list, W represents the sequence of words w(n) induced by A016777. The list includes five partitions and two permutations of the positive integers.

positions of 1-free words in W: A003462;

positions of 0-free words in W: A134342 (conjectured);

positions in W of words w(n) such that # 0's = # 1's: A342732;

positions in W of words w(n) such that # 0's < # 1's: A342733;

positions in W of words w(n) such that # 0's > # 1's: A342734;

positions in W of words w(n) such that first digit = last digit: A342735;

positions in W of words w(n) such that first digit != last digit: A342736;

length of w(n): A342739;

positions in W of words w(n) that end with 0: A342740;

positions in W of words w(n) that end with 1: A342741;

positions in W of words w(n) such that 1st digit = 0 and last digit 0: A342742;

positions in W of words w(n) such that 1st digit = 0 and last digit 1: A342743;

positions in W of words w(n) such that 1st digit = 1 and last digit 0: A342744;

positions in W of words w(n) such that 1st digit = 1 and last digit 1: A342745;

position in W of n-th positive integer (base 2): A342746;

positions in W of binary complement of w(n): A342747;

sum of digits in w(n): A342748;

number of runs in w(n): A342749;

positions in W of palindromes: A342750;

positions in W of words such that #0's - #1's is odd: A342751;

positions in W of words such that #0's - #1's is even: A342752.

For a guide to related sequences, see A341256.

Links

Table of n, a(n) for n=1..86.

Example

The first twenty words w(n): 0, 1, 10, 00, 11, 110, 01, 100, 111, 010, 1110, 101, 000, 1100, 1111, 011, 1010, 11110, 0110, 1101.

Mathematica

z = 250; s = Table[3 n - 2, {n, 1, z}] (* A016777 *)
t = Complement[Range[Max[s]], s] (* A007494 *)
s1[n_] := Length[Intersection[Range[n - 1], s]];
t1[n_] := n - 1 - s1[n];
Table[s1[n], {n, 1, z}] (* A002264 *)
Table[t1[n], {n, 1, z}] (* A004523 *)
w[1] = {0}; w[t[[1]]] = {1};
w[n_] := If[MemberQ[s, n], Join[{0}, w[s1[n]]], Join[{1}, w[t1[n]]]]
tt = Table[w[n], {n, 1, z}] (* A341334, all binary words *)
Flatten[tt] (* A341334, words concatenated *)
Flatten[Position[Map[Union, tt], {0}]] (* A003462 *)
Flatten[Position[Map[Union, tt], {1}]]  (* A134342 conjectured *)
zz = Range[Length[tt]];
Select[zz, Count[tt[[#]], 0] == Count[tt[[#]], 1] &] (* A342732 *)
Select[zz, Count[tt[[#]], 0] < Count[tt[[#]], 1] &]  (* A342733 *)
Select[zz, Count[tt[[#]], 0] > Count[tt[[#]], 1] &]  (* A342734 *)
Select[zz, First[tt[[#]]] == Last[tt[[#]]] &] (* A342735 *)
Select[zz, First[tt[[#]]] != Last[tt[[#]]] &] (* A342736 *)
Map[Length, tt] (* A342739 *)
Select[zz, Last[tt[[#]]] == 0 &] (* A342740 *)
Select[zz, Last[tt[[#]]] == 1 &]  (* A342741 *)
Select[zz, First[tt[[#]]] == 0 && Last[tt[[#]]] == 0 &] (* A342742 *)
Select[zz, First[tt[[#]]] == 0 && Last[tt[[#]]] == 1 &] (* A342743 *)
Select[zz, First[tt[[#]]] == 1 && Last[tt[[#]]] == 0 &] (* A342744 *)
Select[zz, First[tt[[#]]] == 1 && Last[tt[[#]]] == 1 &] (* A342745 *)
d[n_] := If[First[w[n]] == 1, FromDigits[w[n], 2]];
Flatten[Table[Position[Table[d[n], {n, 1, 200}], n], {n, 1, 200}]] (* A342746 *)
comp = Flatten[Table[Position[tt, 1 - w[n]], {n, 1, 50}]] (* A342747 *)
Table[Total[w[n]], {n, 1, 100}]  (* A342748 *)
Map[Length, Table[Map[Length, Split[w[n]]], {n, 1, 100}]] (* A342749 *)
Select[zz, tt[[#]] == Reverse[tt[[#]]] &] (* A342750 *)
Select[zz, OddQ[Count[w[#], 0] - Count[w[#], 1]] &]  (* A342751 *)
Select[zz, EvenQ[Count[w[#], 0] - Count[w[#], 1]] &] (* A342752 *)

Crossrefs

Cf. A016777, A007494, A134352 (conjectured), A341256.

Sequence in context: A366255 A057212 A023959 * A076182 A010058 A140591

Adjacent sequences: A341331 A341332 A341333 * A341335 A341336 A341337

Keyword

nonn

Author

Clark Kimberling, Mar 20 2021

Status

approved