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A350346
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Binary numbers such that when read from right to left, the number of 0's never exceeds the number of 1's.
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2
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0, 1, 11, 101, 111, 1011, 1101, 1111, 10011, 10101, 10111, 11011, 11101, 11111, 100111, 101011, 101101, 101111, 110011, 110101, 110111, 111011, 111101, 111111, 1000111, 1001011, 1001101, 1001111, 1010011, 1010101, 1010111, 1011011, 1011101, 1011111, 1100111
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OFFSET
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1,3
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COMMENTS
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Dyck language interpreted as binary numbers in ascending order (inverse encode of A063171).
The first term a(1)=0 corresponds to an empty string, denote it NULL.
Restoring the leading 0's (need the same number of 0's and 1's) and then replacing "0" by the left parenthesis "(" and "1" by the right parenthesis ")" give well-formed parenthesis strings: 0 -> NULL, 1=01 -> (), 11=0011 -> (()), 101=0101 -> ()(), 111=000111 -> ((())), 1011=001011 -> (()()), 1101=001101 -> (())() and so on.
Chomsky-2 grammar with axiom s, terminal alphabet {0, 1} and three rules s -> ss, s -> 0s1, s -> 01 (compare A063171).
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REFERENCES
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Gennady Eremin, Dyck Numbers, IV. Nested patterns in OEIS A036991, arXiv:2306.10318, 2023.
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LINKS
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FORMULA
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EXAMPLE
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s -> ss -> 0s1s -> 00s11s -> 000111s -> 00011101 = 11101.
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MATHEMATICA
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Join[{0}, Select[Table[FromDigits[IntegerDigits[n, 2]], {n, 120}], Min[Accumulate[ Reverse[ IntegerDigits[#]]/.(0->-1)]]>=0&]] (* Harvey P. Dale, Apr 29 2022 *)
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PROG
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(Python)
def ok(n):
if n == 0: return True
count = {"0": 0, "1": 0}
for bit in bin(n)[:1:-1]:
count[bit] += 1
if count["0"] > count["1"]: return False
nn = 1; print(1, 0)
for n in range(1, 23230): # printing b-file
if ok(n) == False: continue
nn += 1; print(nn, bin(n)[2:])
(Python)
from itertools import count, islice
def A350346_gen(): # generator of terms
yield 0
for n in count(1):
s = bin(n)[2:]
c, l = 0, len(s)
for i in range(l):
c += int(s[l-i-1])
if 2*c <= i:
break
else:
yield int(s)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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