|
| |
|
|
A014081
|
|
a(n) = number of occurrences of '11' in binary expansion of n.
|
|
19
|
|
|
|
0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 1, 2, 3, 0, 0, 0, 1, 0, 0, 1, 2, 1, 1, 1, 2, 2, 2, 3, 4, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 3, 2, 2, 2, 3, 3, 3, 4, 5, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 1, 2, 3, 0, 0, 0, 1, 0, 0, 1, 2, 1, 1, 1, 2, 2, 2, 3, 4, 1, 1, 1, 2, 1, 1, 2, 3, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,8
|
|
|
COMMENTS
|
First occurrence of k: 0, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, ..., A000225[k-1]. [From Robert G. Wilson v, Apr 02 2009]
a(n) = A213629(n,3) for n > 2. - Reinhard Zumkeller, Jun 17 2012
|
|
|
LINKS
|
_Reinhard Zumkeller_, Table of n, a(n) for n = 0..10000
R. Stephan, Some divide-and-conquer sequences ...
R. Stephan, Table of generating functions
Eric Weisstein's World of Mathematics, Digit Block
Eric Weisstein's World of Mathematics, Rudin-Shapiro Sequence
Index entries for sequences related to binary expansion of n
|
|
|
FORMULA
|
a(4n) = a(4n+1) = a(n), a(4n+2) = a(2n+1), a(4n+3) = a(2n+1) + 1. - Ralf Stephan, Aug 21 2003
G.f.: 1/(1-x) * sum(k>=0, t^3/(1+t)/(1+t^2), t=x^2^k). - Ralf Stephan, Sep 10 2003
a(n) = A000120(n) - A069010(n). - Ralf Stephan, Sep 10 2003
|
|
|
EXAMPLE
|
The binary expansion of 15 is 1111, which contains three occurrences of 11, so a(15)=3.
|
|
|
MAPLE
|
# To count occurrences of 11..1 (k times) in binary expansion of v:
cn := proc(v, k) local n, s, nn, i, j, som, kk; som := 0;
kk := cat(seq(1, j = 1 .. k)); n := convert(v, binary);
s := convert(n, string); nn := length(s); for i to nn - k + 1 do
if substring(s, i .. i + k - 1) = kk then som := som + 1 fi
od; RETURN(som) end;
|
|
|
MATHEMATICA
|
f[n_] := Count[ Partition[ IntegerDigits[n, 2], 2, 1], {1, 1}]; Table[ f@n, {n, 0, 104}] [From Robert G. Wilson v, Apr 02 2009]
|
|
|
PROG
|
(Haskell)
import Data.Bits ((.&.))
a014081 n = a000120 (n .&. div n 2) -- Reinhard Zumkeller, Jan 23 2012
(PARI) A014081(n)=sum(i=0, #binary(n)-2, bitand(n>>i, 3)==3) \\ - M. F. Hasler, Jun 06 2012
|
|
|
CROSSREFS
|
Cf. A014082, A033264, A037800, A056973.
Sequence in context: A129753 A147693 A070936 * A091890 A029431 A091492
Adjacent sequences: A014078 A014079 A014080 * A014082 A014083 A014084
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
Simon Plouffe
|
|
|
STATUS
|
approved
|
| |
|
|
