A087116 Number of maximal groups of consecutive zeros in binary representation of n.

1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 2, 2, 2, 2, 1, 2, 2, 3, 2, 3, 3, 3, 2, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 2, 1, 2, 2

Offset: 0

Author

Reinhard Zumkeller, Aug 14 2003

Keywords

• nonn

Comments

The following four statements are equivalent: a(n) = 0; n = 2^k - 1 for some k; A087117(n) = 0; A023416(n) = 0.

Examples

			G.f. = 1 + x^2 + x^4 + x^5 + x^6 + x^8 + x^9 + 2*x^10 + x^11 + x^12 + x^13 + x^14 + ...
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A087118, A087119, A087120, A023416, A007088.

Cf. A023416, A087117.

Cf. A033264, A003817, A069010.

Essentially the same as A033264.

Programs

Haskell

a087116 0 = 1
a087116 n = f 0 n where
   f y 0 = y
   f y x = if r == 0 then g x' else f y x'
           where (x', r) = divMod x 2
                 g z = if r == 0 then g z' else f (y + 1) z'
                       where (z', r) = divMod z 2
-- Reinhard Zumkeller, Mar 31 2015

Mathematica

a[n_] := SequenceCount[IntegerDigits[n, 2], {Longest[0..]}];
Table[a[n], {n, 0, 101}] (* Jean-François Alcover, Oct 18 2021 *)

PARI

a(n) = if (n == 0, 1, hammingweight(bitxor(n, n>>1)) >> 1);
vector(102, i, a(i-1))  \\ Gheorghe Coserea, Sep 17 2015

Python

def A087116(n):
    return sum(1 for d in bin(n)[2:].split('1') if len(d)) # Chai Wah Wu, Nov 04 2016

Formula

a(n) = A033264(n) for n > 0 since strings of 0's alternate with strings of 1's. - Jonathan Sondow, Jan 17 2016

a(n) = a(2*n + 1) = a(4*n + 2) - 1, if n > 0. - Michael Somos, Nov 04 2016

a(n) = A069010(A003817(n)-n) for n > 0. - Chai Wah Wu, Nov 04 2016