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A080116
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Characteristic function of A014486. a(n) = 1 if n's binary expansion is totally balanced, otherwise zero.
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6
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1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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0,1
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COMMENTS
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a(n) = 1 if the binary representation of n forms a valid Dyck path, or equally, a well-formed parenthesization when 1's are converted to left and 0's to right parentheses (that is, when A007088(n) is in A063171), and 0 otherwise. - Antti Karttunen, Aug 23 2019
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LINKS
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EXAMPLE
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0 stands for an empty parenthesization, thus a(0) = 1.
2 has binary expansion "10", which corresponds with "()", thus a(2) = 1.
3 has binary expansion "11", but "((" is not a well-formed parenthesization, thus a(3) = 0.
10 has binary expansion "1010", corresponding with a well-formed parenthesization "()()", thus a(10) = 1.
38 has binary expansion "100110", but "())(()" is not a well-formed parenthesization, thus a(38) = 0.
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MAPLE
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A080116 := proc(n) local c, lev; lev := 0; c := n; while(c > 0) do lev := lev + (-1)^c; c := floor(c/2); if(lev < 0) then RETURN(0); fi; od; if(lev > 0) then RETURN(0); else RETURN(1); fi; end;
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MATHEMATICA
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A080116[n_] := (lev = 0; c = n; While[c > 0, lev = lev + (-1)^c; c = Floor[c/2]; If[lev < 0, Return[0]]]; If[lev > 0, Return[0], Return[1]]); Table[A080116[n], {n, 0, 104}] (* Jean-François Alcover, Jul 24 2013, translated from Maple *)
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PROG
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(Sage)
lev = 0
while n > 0 :
lev += (-1)^n
if lev < 0: return 0
n = n//2
return 0 if lev > 0 else 1
(PARI) A080116(n) = { my(k=0); while(n, k += (-1)^n; n >>= 1; if(k<0, return(0))); (0==k); }; \\ Antti Karttunen, Aug 23 2019
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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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EXTENSIONS
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STATUS
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approved
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