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 A063171 Dyck language interpreted as binary numbers in ascending order. 133
 0, 10, 1010, 1100, 101010, 101100, 110010, 110100, 111000, 10101010, 10101100, 10110010, 10110100, 10111000, 11001010, 11001100, 11010010, 11010100, 11011000, 11100010, 11100100, 11101000, 11110000, 1010101010, 1010101100, 1010110010, 1010110100, 1010111000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n) is the binary expansion of A014486(n). - Joerg Arndt, Feb 27 2013 Replacing "1" by "(" and "0" by ")" yields well-formed bracket expressions (the first term being the empty string) , (), ()(), (()), ()()(), ()(()), (())(), (()()), ((())), ()()()(), ()()(()), ()(())(), ()(()()), ()((())), (())()(), (())(()), (()())(), (()()()), (()(())), ((()))(), ((())()), ((()())), (((()))), ()()()()(), ()()()(()), ()()(())(), ()()(()()), ()()((())), ()(())()(), ()(())(()), ()(()())(), ()(()()()), ()(()(())), ()((()))(), ()((())()), ()((()())), ()(((()))), (())()()(), (())()(()), (())(())(), (())(()()), (())((())), (()())()(), (()())(()), (()()())(), (()()()()), (()()(())), (()(()))(), (()(())()), (()(()())), (()((()))), ((()))()(), ((()))(()), ((())())(), ((())()()), ((())(())), ((()()))(), ((()())()), ((()()())), ((()(()))), (((())))(), (((()))()), (((())())), (((()()))), ((((())))) The term a(0)=0 stands for the empty string. - Joerg Arndt, Feb 27 2013 (Which is actually a leading 0, shown only for zero so that it has a visible representation.) - Daniel Forgues, Feb 27 2013 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..2055 Gennady Eremin, Dynamics of balanced parentheses, lexicographic series and Dyck polynomials, arXiv:1909.07675 [math.CO], 2019. FindStat - Combinatorial Statistic Finder, Dyck paths A. Karttunen, Illustration of initial terms up to size n=7 A. Karttunen, Catalan Automorphisms R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016. Indranil Ghosh, Python program for computing this sequence FORMULA Chomsky-2 grammar with axiom s, terminal alphabet {0, 1} and three rules s -> ss, s -> 1s0, s ->10. a(n) = A071152(n)/2. a(n) = A007088(A014486(n)). EXAMPLE s -> ss -> 1s0s -> 11s00s -> 111000s -> 11100010 MATHEMATICA balancedQ[0] = True; balancedQ[n_] := (s = 0; Do[s += If[b == 1, 1, -1]; If[s < 0, Return[False]], {b, IntegerDigits[n, 2]}]; Return[s == 0]); FromDigits /@ IntegerDigits[ Select[Range[0, 684], balancedQ], 2] (* Jean-François Alcover, Jul 24 2013 *) PROG (Haskell) import Data.Set (singleton, deleteFindMin, union, fromList) newtype Word = Word String deriving (Eq, Show, Read) instance Ord Word where Word us <= Word vs | length us == length vs = us <= vs | otherwise = length us <= length vs a063171 n = a063171_list !! (n-1) a063171_list = dyck \$ singleton (Word "S") where dyck s | null ws = (read w :: Integer) : dyck s' | otherwise = dyck \$ union s' (fromList \$ concatMap gen ws) where ws = filter ((== 'S') . head . snd) \$ map (`splitAt` w) [0..length w - 1] (Word w, s') = deleteFindMin s gen (us, vs) = map (Word . (us ++) . (++ tail vs)) ["10", "1S0", "SS"] -- Reinhard Zumkeller, Mar 09 2011 (Python) def A063171_list(limit): return [0] + [int(bin(k)[2::]) for k in range(1, limit) if is_A014486(k)] print(A063171_list(700)) # Peter Luschny, Jul 30 2022 (Python) from itertools import count, islice from sympy.utilities.iterables import multiset_permutations def A063171_gen(): # generator of terms yield 0 for l in count(1): for s in multiset_permutations('0'*l+'1'*(l-1)): c, m = 0, (l<<1)-1 for i in range(m): if s[i] == '1': c += 2 if c

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Last modified February 21 03:13 EST 2024. Contains 370219 sequences. (Running on oeis4.)