A342910 Concatenation of all 01-words, in the order induced by A032766; see Comments.

0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 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 = A032766; t = A016789; s' = A004396; t' = A002264;

In the following list, W represents the sequence of words w(n) induced by A032766. The list includes five partitions and a self-inverse permutation of the positive integers.

length of w(n): A344150;

positions in W of words w(n) such that # 0's = # 1's: A344151;

positions in W of words w(n) such that # 0's < # 1's: A344152;

positions in W of words w(n) such that # 0's > # 1's: A344153;

positions in W of words w(n) that end with 0: A344154;

positions in W of words w(n) that end with 1: A344155;

positions in W of words w(n) such that first digit = last digit: A344156;

positions in W of words w(n) such that first digit != last digit: A344157;

positions in W of words w(n) such that 1st digit = 0 and last digit 0: A344158;

positions in W of words w(n) such that 1st digit = 0 and last digit 1: A344159;

positions in W of words w(n) such that 1st digit = 1 and last digit 0: A344160;

positions in W of words w(n) such that 1st digit = 1 and last digit 1: A344161;

position in W of n-th positive integer (base 2): A344162;

positions in W of binary complement of w(n): A344163;

sum of digits in w(n): A344164;

number of runs in w(n): A344165;

positions in W of palindromes: A344166;

positions in W of words such that #0's - #1's is odd: A344167;

positions in W of words such that #0's - #1's is even: A344168;

positions in W of the reversal of the n-th word in W: A344169.

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, 00, 01, 10, 000, 001, 11, 010, 0000, 100, 0001, 011, 101, 0010, 00000, 110, 0100, 00001, 1000.

Mathematica

z = 250;
"The sequence s:"  (* A001651, (3n/2) *)
s = Table[Floor[3 n/2], {n, 1, z}]
"The sequence t:" (* A016789; congr to 0 or 1 mod 3;  *)
t = Complement[Range[Max[s]], s]
s1[n_] := Length[Intersection[Range[n - 1], s]];
t1[n_] := n - 1 - s1[n];
"The sequence s1:"
Table[s1[n], {n, 1, z}] (* A004396 *)
"The sequence t1:"
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]]]]
"List tt of all binary words:"
tt = Table[w[n], {n, 1, z}] (*  all the binary words *)
"All the words, concatenated:"
Flatten[tt] (* words, concatenated, A344150 *)
"Positions of words in which #0's = #1's:"  (* A344151 *)
Select[Range[Length[tt]], Count[tt[[#]], 0] == Count[tt[[#]], 1] &]
"Positions of words in which #0's < #1's:"  (* A344152 *)
Select[Range[Length[tt]], Count[tt[[#]], 0] < Count[tt[[#]], 1] &]
"Positions of words in which #0's > #1's:"  (* A344153 *)
Select[Range[Length[tt]], Count[tt[[#]], 0] > Count[tt[[#]], 1] &]
"Positions of words ending with 0:" (* A344154 *)
Select[Range[Length[tt]], Last[tt[[#]]] == 0 &]
"Positions of words ending with 1:" (* A344155 *)
Select[Range[Length[tt]], Last[tt[[#]]] == 1 &]
"Positions of words starting and ending with same digit:" (* A344156 *)
Select[Range[Length[tt]], First[tt[[#]]] == Last[tt[[#]]] &]
"Positions of words starting and ending with opposite digits:" (* A344157 *)
Select[Range[Length[tt]], First[tt[[#]]] != Last[tt[[#]]] &]
"Positions of words starting with 0 and ending with 0:" (* A344158 *)
Select[Range[Length[tt]], First[tt[[#]]] == 0 && Last[tt[[#]]] == 0 &]
"Positions of words starting with 0 and ending with 1:" (* A344159 *)
Select[Range[Length[tt]], First[tt[[#]]] == 0 && Last[tt[[#]]] == 1 &]
"Positions of words starting with 1 and ending with 0:" (* A344160 *)
Select[Range[Length[tt]], First[tt[[#]]] == 1 && Last[tt[[#]]] == 0 &]
"Positions of words starting with 1 and ending with 1:" (* A344161 *)
Select[Range[Length[tt]], First[tt[[#]]] == 1 && Last[tt[[#]]] == 1 &]
"Position of n-th positive integer (base 2) in tt:  A344162 "
d[n_] := If[First[w[n]] == 1, FromDigits[w[n], 2]];
Flatten[Table[Position[Table[d[n], {n, 1, 200}], n], {n, 1, 200}]]
"Position of binary complement of w(n): A344163"
comp = Flatten[Table[Position[tt, 1 - w[n]], {n, 1, 50}]]
"Sum of digits of w(n):  A344164"
Table[Total[w[n]], {n, 1, 100}]
"Number of runs in w(n):  A344165"
Map[Length, Table[Map[Length, Split[w[n]]], {n, 1, 100}]]
"Palindromes:"
Select[tt, # == Reverse[#] &]
"Positions of palindromes:  A344166"
Select[Range[Length[tt]], tt[[#]] == Reverse[tt[[#]]] &]
"Positions of words in which #0's - #1's is odd:  A344167"
Select[Range[Length[tt]], OddQ[Count[w[#], 0] - Count[w[#], 1]] &]
"Positions of words in which #0's - #1's is even: A344168"
Select[Range[Length[tt]], EvenQ[Count[w[#], 0] - Count[w[#], 1]] &]
"Position of the reversal of the n-th word:  A344169"
Flatten[Table[Position[tt, Reverse[w[n]]], {n, 1, 150}]]

Crossrefs

Cf. A032766, A016789, A004396, A002264, A341256.

Sequence in context: A284751 A288426 A286052 * A341258 A285831 A188294

Adjacent sequences: A342907 A342908 A342909 * A342911 A342912 A342913

Keyword

nonn,base

Author

Clark Kimberling, May 11 2021

Status

approved