A341258 Concatenation of all 01-words, in the order induced by A000201; see Comments.

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

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 = A000201; t = A001950; s' = A005206; t' = A060144;

For a guide to related sequences, see the Mathematica program and A341256.

Links

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

Example

The first 20 words: 0,1,00,01,10,000,11,001,010,100,0000,011,101,0001,110,0010,0100,1000,00000,111.

Mathematica

z = 250; r = GoldenRatio;
"The sequence s; "  (* A000201 *)
s = Table[Floor[n r], {n, 1, z}]
"The sequence t:" (* A001950  *)
t = Complement[Range[Max[s]], s]
s1[n_] := Length[Intersection[Range[n - 1], s]];
t1[n_] := n - 1 - s1[n];
"The sequence s1: A005206"
Table[s1[n], {n, 1, z}]
"The sequence t1: A060144"
Table[t1[n], {n, 1, z}]
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] (* A341258  words, concatenated *)
"Map of Union onto the words:"
Map[Union, tt]
"Length of w[n]: A112310"
Map[Length, tt]
"Positions of words in which #0's = #1's: A344950"
"This and the next two sequences partition N."
Select[Range[Length[tt]],
Count[tt[[#]], 0] == Count[tt[[#]], 1] &]
"Positions of words in which #0's < #1's: A344951"
Select[Range[Length[tt]], Count[tt[[#]], 0] < Count[tt[[#]], 1] &]
"Positions of words in which #0's > #1's: A344952"
Select[Range[Length[tt]], Count[tt[[#]], 0] > Count[tt[[#]], 1] &]
"Positions of words ending with 0:  A133512 send comment"
Select[Range[Length[tt]], Last[tt[[#]]] == 0 &]
"Positions of words ending with 1: A344953"
Select[Range[Length[tt]], Last[tt[[#]]] == 1 &]
"Positions of words starting and ending with same digit: A344954"
Select[Range[Length[tt]], First[tt[[#]]] == Last[tt[[#]]] &]
"Positions of words starting and ending with opposite digits: A344955"
Select[Range[Length[tt]], First[tt[[#]]] != Last[tt[[#]]] &]
"Positions of words starting with 0 and ending with 0: A344956"
Select[Range[Length[tt]],
First[tt[[#]]] == 0 && Last[tt[[#]]] == 0 &]
"Positions of words starting with 0 and ending with 1: A344957"
Select[Range[Length[tt]],
First[tt[[#]]] == 0 && Last[tt[[#]]] == 1 &]
"Positions of words starting with 1 and ending with 0:  A344958"
Select[Range[Length[tt]],
First[tt[[#]]] == 1 && Last[tt[[#]]] == 0 &]
"Positions of words starting with 1 and ending with 1: A344959"
Select[Range[Length[tt]],
First[tt[[#]]] == 1 && Last[tt[[#]]] == 1 &]
"Position of n-th positive integer (base 2) in tt:  A344988"
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): A344960"
comp = Flatten[Table[Position[tt, 1 - w[n]], {n, 1, 100}]]
"Sum of digits of w(n):  A206650"
Table[Total[w[n]], {n, 1, 100}]
"Number of runs in w(n):  A344961"
Map[Length, Table[Map[Length, Split[w[n]]], {n, 1, 100}]]
"Palindromes:"
Select[tt, # == Reverse[#] &]
"Positions of palindromes:  A341333"
Select[Range[Length[tt]], tt[[#]] == Reverse[tt[[#]]] &]
"Positions of words in which #0's - #1's is odd:  A095879"
Select[Range[Length[tt]],
OddQ[Count[w[#], 0] - Count[w[#], 1]] &]
"Positions of words in which #0's - #1's is even: A095880"
Select[Range[Length[tt]], EvenQ[Count[w[#], 0] - Count[w[#], 1]] &]
"Position of the reversal of the n-th word:  A344962"
u21 = Flatten[Table[Position[tt, Reverse[w[n]]], {n, 1, 150}]]

Crossrefs

Cf. A000201, A001950, A005206, A060144, A112310, A341259, A341333.

Sequence in context: A288426 A286052 A342910 * A285831 A188294 A079101

Adjacent sequences: A341255 A341256 A341257 * A341259 A341260 A341261

Keyword

nonn

Author

Clark Kimberling, Mar 16 2021

Status

approved