A056974 Number of blocks of {0, 0, 0} in the binary expansion of n.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 2, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 3, 2, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 2, 1, 1, 0, 0, 0

Offset

1,16

Comments

Overlaps count. For example, 64 in binary is 1000000, which means that a(64) = 4, not 2. - Harvey P. Dale, Jan 10 2016

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Eric Weisstein's World of Mathematics, Digit Block

Index entries for sequences related to binary expansion of n

Formula

a(1) = 0, and then after, a(2n) = a(n) + [n congruent to 0 mod 8], a(2n+1) = a(n). - Ralf Stephan, Aug 22 2003, corrected by Antti Karttunen, Oct 10 2017

Mathematica

a[n_, bits_] := (idn = IntegerDigits[n, 2]; ln = Length[idn]; lb = Length[bits]; For[cnt = 0; k = 1, k <= ln - lb + 1, k++, If[idn[[k ;; k + lb - 1]] == bits, cnt++]]; cnt); Table[ a[n, {0, 0, 0}], {n, 1, 102} ] (* Jean-François Alcover, Oct 23 2012 *)
Table[SequenceCount[IntegerDigits[n, 2], {0, 0, 0}, Overlaps->True], {n, 110}] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, Jan 10 2016 *)

Prog

(PARI) a(n)=my(v=binary(n)); sum(i=3, #v, v[i]+v[i-1]+v[i-2]==0) \\ Charles R Greathouse IV, Dec 07 2011
(PARI)
a(n) = {
  my(x = bitor(n, bitor(n>>1, n>>2)));
  if (x == 0, 0, 1 + logint(x, 2) - hammingweight(x))
};
vector(102, i, a(i))  \\ Gheorghe Coserea, Sep 17 2015
(Scheme)
;; This uses Ralf Stephan's recurrence and memoization-macro definec:
(definec (A056974 n) (cond ((= 1 n) 0) ((even? n) (+ (if (zero? (modulo n 8)) 1 0) (A056974 (/ n 2)))) (else (A056974 (/ (- n 1) 2))))) ;; Antti Karttunen, Oct 10 2017

Crossrefs

Cf. A014082, A056974, A056975, A056976, A056977, A056978, A056979, A056980.

Sequence in context: A010109 A304021 A378357 * A368844 A377336 A181563

Adjacent sequences: A056971 A056972 A056973 * A056975 A056976 A056977

Keyword

nonn,base,easy

Author

Eric W. Weisstein

Status

approved