A014083 Occurrences of '1111' in binary expansion of n.

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

Offset

0,32

Links

Antti Karttunen, Table of n, a(n) for n = 0..65536

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

R. Stephan, Table of generating functions

Index entries for sequences related to binary expansion of n

Formula

a(2n) = a(n), a(2n+1) = a(n) + [n congruent to 7 mod 8]. - Ralf Stephan, Aug 21 2003

G.f.: 1/(1-x) * sum(k>=0, t^15(1-t)/(1-t^16), t=x^2^k). - Ralf Stephan, Sep 08 2003

a(n) <= log_2(n+1) - 3 for n >= 7. - Charles R Greathouse IV, Jan 21 2016

Example

a(63) = 3 as 63 = 111111 in binary and 1111 occurs three times (different occurrences may overlap). - Antti Karttunen, Jul 24 2017

Maple

See A014081.

Mathematica

Table[SequenceCount[IntegerDigits[n, 2], {1, 1, 1, 1}, Overlaps->True], {n, 0, 100}] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, Sep 25 2015 *)

Prog

(PARI) u1111(n)=my(v=binary(n)); sum(k=1, #v-3, v[k]&&v[k+1]&&v[k+2]&&v[k+3])
(PARI) a(n)=my(s, t); while(n, n>>=valuation(n, 2); t=valuation(n+1, 2); s+=max(t-3, 0); n>>=t); s \\ Charles R Greathouse IV, Jan 21 2016

Crossrefs

Cf. A000120, A014081, A014082.

Sequence in context: A228594 A281669 A331845 * A238403 A112315 A295663

Adjacent sequences: A014080 A014081 A014082 * A014084 A014085 A014086

Keyword

nonn,base,easy

Author

Simon Plouffe

Extensions

Term a(0)=0 prepended and more terms from Antti Karttunen, Jul 24 2017

Status

approved