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A080116 Characteristic function of A014486. a(n) = 1 if n's binary expansion is totally balanced, otherwise zero. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..65537

Index entries for sequences related to binary expansion of n

Index entries for characteristic functions

Index entries for sequences related to parenthesizing

EXAMPLE

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.

MAPLE

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;

MATHEMATICA

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-Fran├žois Alcover, Jul 24 2013, translated from Maple *)

PROG

(Sage)

def A080116(n) :

    lev = 0

    while n > 0 :

        lev += (-1)^n

        if lev < 0: return 0

        n = n//2

    return 0 if lev > 0 else 1

[A080116(n) for n in (0..104)] # Peter Luschny, Aug 09 2012

(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

CROSSREFS

Cf. A014486, A063171, A080110, A080111, A080300, A080301.

Sequence in context: A016378 A323513 A089805 * A016359 A014029 A016396

Adjacent sequences:  A080113 A080114 A080115 * A080117 A080118 A080119

KEYWORD

nonn,base

AUTHOR

Antti Karttunen Feb 11 2003

EXTENSIONS

Examples added by Antti Karttunen, Aug 23 2019

STATUS

approved

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Last modified November 20 14:54 EST 2019. Contains 329337 sequences. (Running on oeis4.)