A080110 -id:A080110 - cp's OEIS frontend

A080116 Characteristic function of A014486. a(n) = 1 if n's binary expansion is totally balanced, otherwise zero.

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

Offset: 0

Antti Karttunen, Feb 11 2003

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

			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.

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

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;

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 *)

A080116(n) = { my(k=0); while(n, k += (-1)^n; n >>= 1; if(k<0, return(0))); (0==k); }; \\ Antti Karttunen, Aug 23 2019

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

Examples added by Antti Karttunen, Aug 23 2019

A080112 Positions of A080114 in A000040.

Original entry on oeis.org

2, 3, 4, 5, 6, 9, 11, 12, 15, 17, 20, 22, 23, 27, 32, 36, 39, 43, 46, 52, 54, 56, 58, 64, 72, 76, 81, 83, 85, 92, 96, 103, 109, 111, 118, 120, 128, 132, 133, 146, 150, 154, 156, 157, 164, 166, 167, 173, 175, 179, 182, 185, 190, 200, 202, 207, 215, 222, 225, 228, 229

Offset: 1

Antti Karttunen, Feb 11 2003

nonn

Complement of A080113. Characteristic function: A080110.

A080112 := proc(n) option remember; local i; if(1 = n) then RETURN(2); fi; i := A080112(n-1)+1; while(i > 0) do if(A080110(i) > 0) then RETURN(i); fi; i := i+1; od; end;

s[p_, u_] := Sum[JacobiSymbol[j, p], {j, 1, u}]; Select[Range[2, 300], (p = Prime[#]; AllTrue[Range[(p - 1)/2], s[p, #] >= 0 &]) &] (* Jean-François Alcover, Mar 07 2016 *)

@CachedFunction
def A080112(n) :
     if 1 == n : return 2
     i = A080112(n-1) + 1
     while i > 0 :
         if A080110(i) > 0 : return i
         i += 1
[A080112(n) for n in (1..61)] # Peter Luschny, Aug 09 2012

cp's OEIS Frontend

A080116 Characteristic function of A014486. a(n) = 1 if n's binary expansion is totally balanced, otherwise zero.

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