A362240 Triangle read by rows: Row n is the shortest, then lexicographically earliest sequence of 0s and 1s not yet in the sequence.

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

Offset

1

Comments

Bit strings as indexed by A362009.

Shortest then lexicographically earliest ordering for rows is as in A076478.

Links

Samuel Harkness, Table of n, a(n) for n = 1..10008 (first 911 rows flattened)

Neal Gersh Tolunsky, Bitmap of a(n), n = 1..2^17, read left to right, top to bottom, where black represents 1 and white 0.

Samuel Harkness, MATLAB program

Example

The triangle begins:
   n      row(n)
  ---     ------
   1      0;
   2      1;
   3      0, 0;
   4      1, 1;
   5      0, 0, 0;
   6      1, 0, 1;
   7      1, 1, 1;
   8      0, 0, 0, 0;
   9      1, 0, 1, 0;
  10      1, 1, 0, 1;
  11      0, 0, 0, 0, 0;
  12      0, 0, 0, 1, 1;
  13      0, 0, 1, 0, 0;
For row 6, we know that all binary sequences up to row 5 = {0, 0, 0} appear, so we first check {0, 0, 1}. This sequence appears at {a(3), a(4), a(5)} (using parts of rows 3 and 4).
Next, check {0, 1, 0}. This sequence appears at {a(1), a(2), a(3)} (using parts of rows 1, 2, and 3).
Next, check {0, 1, 1}. This sequence appears at {a(4), a(5), a(6)} (using parts of rows 3 and 4).
Next, check {1, 0, 0}. This sequence appears at {a(2), a(3), a(4)} (using parts of rows 2 and 3).
Next, check {1, 0, 1}. This sequence does not appear anywhere in the sequence, so row 6 of the triangle is {1, 0, 1}.

Mathematica

V = {0}; K = {0}; While[Length@K <= 87, y = 0; While[y == 0, i = Length@V; V[[i]]++; While[V[[i]] == 2 && i > 1 , V[[i]] = 0; i--; V[[i]]++]; If[V[[1]] == 2, V = ConstantArray[0, Length@V + 1]]; z = 0; For[a = 1, a <= Length@K - Length@V + 1, a++, If[K[[a ;; a + Length@V - 1]] == V, z = 1; Break[]]]; If[z == 0, AppendTo[K, V]; K = Flatten[K]; y = 1]]]; Print[K[[1 ;; 87]]]

Prog

(MATLAB) See Links section.
(Python)
from itertools import chain, count, islice, product
def bins(): yield from ("".join(b) for d in count(1) for b in product("01", repeat=d))
def agen(s=""): yield from chain.from_iterable(map(int, t) for t in bins() if t not in s and (s:=s+t))
print(list(islice(agen(), 87))) # Michael S. Branicky, Apr 12 2023

Crossrefs

Cf. A076478, A362009, A362241 (rows interpreted as binary).

Cf. A118247

Sequence in context: A079944 A059652 A108736 * A079813 A078580 A059651

Adjacent sequences: A362237 A362238 A362239 * A362241 A362242 A362243

Keyword

nonn,tabf

Author

Samuel Harkness, Apr 12 2023

Status

approved