A014082 Number of occurrences of '111' in binary expansion of n.

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

Offset

0,16

Comments

a(n) = A213629(n,7) for n > 6. - Reinhard Zumkeller, Jun 17 2012

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Ralf Stephan, Some divide-and-conquer sequences ...

Ralf Stephan, Table of generating functions

Eric Weisstein's World of Mathematics, Digit Block

Index entries for sequences related to binary expansion of n

Formula

a(2n) = a(n), a(2n+1) = a(n) + [n congruent to 3 mod 4]. - Ralf Stephan, Aug 21 2003

G.f.: 1/(1-x) * Sum_{k>=0} t^7(1-t)/(1-t^8), where t=x^2^k. - Ralf Stephan, Sep 08 2003

Maple

See A014081.
f:= proc(n) option remember;
  if n::even then procname(n/2)
  elif n mod 8 = 7 then 1 + procname((n-1)/2)
  else procname((n-1)/2)
fi
end proc:
f(0):= 0:
map(f, [$0..1000]); # Robert Israel, Sep 11 2015

Mathematica

f[n_] := Count[ Partition[ IntegerDigits[n, 2], 3, 1], {1, 1, 1}]; Table[f@n, {n, 0, 104}] (* Robert G. Wilson v, Apr 02 2009 *)
a[0] = a[1] = 0; a[n_] := a[n] = If[EvenQ[n], a[n/2], a[(n - 1)/2] + Boole[Mod[(n - 1)/2, 4] == 3]]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Oct 22 2012, after Ralf Stephan *)
Table[SequenceCount[IntegerDigits[n, 2], {1, 1, 1}, Overlaps->True], {n, 0, 110}] (* Harvey P. Dale, Mar 05 2023 *)

Prog

(Haskell)
import Data.List (tails, isPrefixOf)
a014082 = sum . map (fromEnum . ([1, 1, 1] `isPrefixOf`)) .
                    tails . a030308_row
-- Reinhard Zumkeller, Jun 17 2012
(PARI) a(n) = hammingweight(bitand(n, bitand(n>>1, n>>2))); \\ Gheorghe Coserea, Aug 30 2015

Crossrefs

Cf. A014081, A033264, A056974, A056975, A056976, A056977, A056978, A056979, A056980, A213629, A239906, A239907.

Sequence in context: A086078 A353462 A323913 * A322583 A351439 A102354

Adjacent sequences: A014079 A014080 A014081 * A014083 A014084 A014085

Keyword

nonn,easy,base

Author

Simon Plouffe

Status

approved