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A341334 Concatenation of all 01-words, in the order induced by A016777; see Comments. 22
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
Clark Kimberling, Mar 20 2021
STATUS
approved

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Last modified March 19 04:26 EDT 2024. Contains 370952 sequences. (Running on oeis4.)