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A014081 a(n) = number of occurrences of '11' in binary expansion of n. 26
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

a(n) takes the value k for the first time at n = 2^(k+1)-1. Cf. A000225. - 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

Helmut Prodinger, Generalizing the sum of digits function, SIAM J. Algebraic Discrete Methods 3 (1982), no. 1, 35--42. MR0644955 (83f:10009).. [See B_2(11,n) on p. 35. - N. J. A. Sloane, Apr 06 2014]

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)), where 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 := convert(cat(seq(1, j = 1 .. k)), string);

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;

som; end; # This program no longer worked. Corrected by N. J. A. Sloane, Apr 06 2014.

[seq(cn(n, 2), n=0..300)];

# Alternative:

A014081 := proc(n) option remember;

  if n mod 4 <= 1 then procname(floor(n/4))

elif n mod 4 = 2 then procname(n/2)

else 1 + procname((n-1)/2)

fi

end proc:

A014081(0):= 0:

map(A014081, [$0..1000]); # Robert Israel, Sep 04 2015

MATHEMATICA

f[n_] := Count[ Partition[ IntegerDigits[n, 2], 2, 1], {1, 1}]; Table[ f@n, {n, 0, 104}] (* 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

(PARI) a(n) = hammingweight(bitand(n, n>>1)) ;

vector(105, i, a(i-1))  \\ Gheorghe Coserea, Aug 30 2015

CROSSREFS

Cf. A014082, A033264, A037800, A056973, A000225, A213629, A000120, A069010.

First differences give A245194.

A245195 gives 2^a(n).

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

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Last modified December 10 11:53 EST 2016. Contains 279001 sequences.