A342753 Concatenation of all 01-words, in the order induced by A001651; see Comments.

0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 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 = A001651; t = A008585; s' = A004523; t' = A002264;

In the following list, W represents the sequence of words w(n) induced by A001651. The list includes five partitions and two permutations of the positive integers.

positions of 1-free words in W: A006999;

positions of 0-free words in W: A029858;

length of w(n): A342774;

positions in W of words w(n) such that # 0's = # 1's: A342775;

positions in W of words w(n) such that # 0's < # 1's: A342776;

positions in W of words w(n) such that # 0's > # 1's: A342777;

positions in W of words having last digit 0: A342778;

positions in W of words having last digit 1: A342779;

positions in W of words w(n) such that first digit = last digit: A342780;

positions in W of words w(n) such that first digit != last digit: A342781;

positions in W of words w(n) such that 1st digit = 0 and last digit 0: A342748;

positions in W of words w(n) such that 1st digit = 0 and last digit 1: A342783;

positions in W of words w(n) such that 1st digit = 1 and last digit 0: A342784;

positions in W of words w(n) such that 1st digit = 1 and last digit 1: A342785;

position in W of n-th positive integer (base 2): A342786;

positions in W of binary complement of w(n): A342787;

sum of digits in w(n): A342788;

number of runs in w(n): A342789;

positions in W of palindromes: A342790;

positions in W of words such that #0's - #1's is odd: A342791;

positions in W of words such that #0's - #1's is even: A342792.

position in W of the reversal of the n-th word in A342798.

For a guide to related sequences, see A341256.

Links

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

Example

The first 16 words w(n): 0, 00, 1, 000, 01, 10, 0000, 001, 100, 010, 00000, 11, 0001, 0100, 1000, 0010.

Mathematica

z = 100; s = Table[Floor[(3 n - 1)/2], {n, 1, z}]; (* A001651 *)
t = Complement[Range[Max[s]], s]; (* A008585 *)
s1[n_] := Length[Intersection[Range[n - 1], s]];
t1[n_] := n - 1 - s1[n];
Table[s1[n], {n, 1, z}]; (* A004523 *)
Table[t1[n], {n, 1, z}]; (* A002264 *)
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}] (* A342753, words *)
Flatten[tt] (* A342753, concatenated *)
Map[Length, tt] (* A342774 *)
Flatten[Position[Map[Union, tt], {0}]]; (* A006999 *)
Flatten[Position[Map[Union, tt], {1}]];  (* A029858 *)
zz = Range[Length[tt]];
Select[zz, Count[tt[[#]], 0] == Count[tt[[#]], 1] &] (* A342775 *)
Select[zz, Count[tt[[#]], 0] < Count[tt[[#]], 1] &] (* A342776 *)
Select[zz, Count[tt[[#]], 0] > Count[tt[[#]], 1] &] (* A342777 *)
Select[zz, Last[tt[[#]]] == 0 &]  (* A342778 *)
Select[zz, Last[tt[[#]]] == 1 &]  (* A342779 *)
Select[zz, First[tt[[#]]] == Last[tt[[#]]] &] (* A342780 *)
Select[zz, First[tt[[#]]] != Last[tt[[#]]] &]  (* A342781 *)
Select[zz, First[tt[[#]]] == 0 && Last[tt[[#]]] == 0 &] (* A342782 *)
Select[zz, First[tt[[#]]] == 0 && Last[tt[[#]]] == 1 &] (* A342783 *)
Select[zz, First[tt[[#]]] == 1 && Last[tt[[#]]] == 0 &] (* A342784 *)
Select[zz, First[tt[[#]]] == 1 && Last[tt[[#]]] == 1 &] (* A342785 *)
d[n_] := If[First[w[n]] == 1, FromDigits[w[n], 2]];
Flatten[Table[Position[Table[d[n], {n, 1, 200}], n], {n, 1, 200}]] (* A342786 *)
comp = Flatten[Table[Position[tt, 1 - w[n]], {n, 1, 50}]]   (* A342787 *)
Table[Total[w[n]], {n, 1, 100}]  (* A342788 *)
Map[Length, Table[Map[Length, Split[w[n]]], {n, 1, 100}]] (* A342789 *)
Select[zz, tt[[#]] == Reverse[tt[[#]]] &] (* A342790 *)
Select[zz, OddQ[Count[w[#], 0] - Count[w[#], 1]] &] (* A342791 *)
Select[zz, EvenQ[Count[w[#], 0] - Count[w[#], 1]] &] (* A342792 *)
Flatten[Table[Position[tt, Reverse[w[n]]], {n, 1, 150}]];  (* A342798 *)

Crossrefs

Cf. A001651, A341256, A342910.

Sequence in context: A359474 A359429 A353470 * A358752 A368913 A354820

Adjacent sequences: A342750 A342751 A342752 * A342754 A342755 A342756

Keyword

nonn,base

Author

Clark Kimberling, Apr 10 2021

Status

approved