A056974 -id:A056974 - cp's OEIS frontend

A033264 Number of blocks of {1,0} in the binary expansion of n.

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

Offset: 1

Clark Kimberling

Number of i such that d(i) < d(i-1), where Sum_{d(i)*2^i: i=0,1,....,m} is base 2 representation of n.

This is the base-2 down-variation sequence; see A297330. - Clark Kimberling, Jan 18 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions.
Eric Weisstein's World of Mathematics, Digit Block.
Index entries for sequences related to binary expansion of n

Cf. A014081, A014082, A037800, A056974, A056975, A056976, A056977, A056978, A056979, A056980, A196521.
a(n) = A005811(n) - ceiling(A005811(n)/2) = A005811(n) - A069010(n).
Equals (A072219(n+1)-1)/2.
Cf. also A175047, A030308.
Essentially the same as A087116.

a033264 = f 0 . a030308_row where
   f c [] = c
   f c (0 : 1 : bs) = f (c + 1) bs
   f c (_ : bs) = f c bs
-- Reinhard Zumkeller, Feb 20 2014, Jun 17 2012

f:= proc(n) option remember; local k;
k:= n mod 4;
if k = 2 then procname((n-2)/4) + 1
elif k = 3 then procname((n-3)/4)
else procname((n-k)/2)
fi
end proc:
f(1):= 0: f(0):= q:
seq(f(i),i=1..100); # Robert Israel, Aug 31 2015

Table[Count[Partition[IntegerDigits[n, 2], 2, 1], {1, 0}], {n, 102}] (* Michael De Vlieger, Aug 31 2015, after Robert G. Wilson v at A014081 *)
Table[SequenceCount[IntegerDigits[n,2],{1,0}],{n,110}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 26 2017 *)

a(n) = { hammingweight(bitand(n>>1, bitneg(n))) }; \\ Gheorghe Coserea, Aug 30 2015

def A033264(n): return ((n>>1)&~n).bit_count() # Chai Wah Wu, Jun 25 2025

G.f.: 1/(1-x) * Sum_(k>=0, t^2/(1+t)/(1+t^2), t=x^2^k). - Ralf Stephan, Sep 10 2003
a(n) = A069010(n) - (n mod 2). - Ralf Stephan, Sep 10 2003
a(4n) = a(4n+1) = a(2n), a(4n+2) = a(n)+1, a(4n+3) = a(n). - Ralf Stephan, Aug 20 2003
a(n) = A087116(n) for n > 0, since strings of 0's alternate with strings of 1's, which end in (1,0). - Jonathan Sondow, Jan 17 2016
Sum_{n>=1} a(n)/(n*(n+1)) = Pi/4 - log(2)/2 (A196521) (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021

A037800 Number of occurrences of 01 in the binary expansion of n.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

Number of i such that d(i)>d(i-1), where Sum{d(i)*2^i: i=0,1,...,m} is base 2 representation of n.

This is the base-2 up-variation sequence; see A297330. - Clark Kimberling, Jan 18 2017

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions.
Eric Weisstein's World of Mathematics, Digit Block.
Index entries for sequences related to binary expansion of n

Cf. A014081, A014082, A033264, A056974, A056975, A056976, A056977, A056978, A056979, A056980.

Cf. A030308, A231902.

a037800 = f 0 . a030308_row where
   f c [_]          = c
   f c (1 : 0 : bs) = f (c + 1) bs
   f c (_ : bs)     = f c bs
-- Reinhard Zumkeller, Feb 20 2014

Table[SequenceCount[IntegerDigits[n,2],{0,1}],{n,0,120}] (* Harvey P. Dale, Aug 10 2023 *)

a(n) = { if(n == 0, 0, -1 + hammingweight(bitnegimply(n, n>>1))) };  \\ Gheorghe Coserea, Aug 31 2015

a(2n) = a(n), a(2n+1) = a(n) + [n is even]. - Ralf Stephan, Aug 21 2003
G.f.: 1/(1-x) * Sum_{k>=0} t^5/(1+t)/(1+t^2) where t=x^2^k. - Ralf Stephan, Sep 10 2003
a(n) = A069010(n) - 1, n>0. - Ralf Stephan, Sep 10 2003
Sum_{n>=1} a(n)/(n*(n+1)) = log(2)/2 + Pi/4 - 1 = A231902 - 1 (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021

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

Original entry on oeis.org

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

Simon Plouffe

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

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

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

import Data.List (tails, isPrefixOf)
a014082 = sum . map (fromEnum . ([1,1,1] `isPrefixOf`)) .
                    tails . a030308_row
-- Reinhard Zumkeller, Jun 17 2012

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

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 *)

a(n) = hammingweight(bitand(n, bitand(n>>1, n>>2))); \\ Gheorghe Coserea, Aug 30 2015

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

A056978 Number of blocks of {1, 0, 0} in binary expansion of n.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein

a(n) = A213629(n,4) for n > 3. - Reinhard Zumkeller, Jun 17 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digit Block

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

import Data.List (tails, isPrefixOf)
a056978 = sum . map (fromEnum . ([0,0,1] `isPrefixOf`)) .
                    tails . a030308_row
-- Reinhard Zumkeller, Jun 17 2012

a[1] = a[2] = 0; a[n_] := a[n] = If[OddQ[n], a[(n-1)/2], a[n/2] + Boole[Mod[n/2, 4] == 2]]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Oct 22 2012, after Ralf Stephan *)
Table[SequenceCount[IntegerDigits[n,2],{1,0,0}],{n,120}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 01 2016 *)

a(n) = hammingweight(bitnegimply(n>>2, bitor(n>>1, n)));  \\ Gheorghe Coserea, Sep 08 2015

a(2n) = a(n) + [n congruent to 2 mod 4], a(2n+1) = a(n). - Ralf Stephan, Aug 22 2003

A056979 Number of blocks of {1, 0, 1} in binary expansion of n.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein

a(n) = A213629(n,5) for n > 4. - Reinhard Zumkeller, Jun 17 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J.-P. Allouche, J. Shallit, The Ring of k-regular Sequences II
Eric Weisstein's World of Mathematics, Digit Block
Index entries for sequences related to binary expansion of n

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

import Data.List (tails, isPrefixOf)
a056979 = sum . map (fromEnum . ([1,0,1] `isPrefixOf`)) .
                    tails . a030308_row
-- Reinhard Zumkeller, Jun 17 2012

a[1] = a[2] = 0; a[n_] := a[n] = If[EvenQ[n], a[n/2], a[(n - 1)/2] + Boole[Mod[(n - 1)/2, 4] == 2]]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Oct 22 2012, after Ralf Stephan *)

a(n) = hammingweight(bitnegimply(bitand(n, n>>2), n>>1));
vector(102, i, a(i))  \\ Gheorghe Coserea, Sep 17 2015

a(2n) = a(n), a(2n+1) = a(n) + [n congruent to 2 mod 4]. - Ralf Stephan, Aug 22 2003

A056980 Number of blocks of {1, 1, 0} in binary expansion of n.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digit Block.

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

Cf. A213629.

import Data.List (tails, isPrefixOf)
a056980 = sum . map (fromEnum . ([0,1,1] `isPrefixOf`)) .
                    tails . a030308_row
-- Reinhard Zumkeller, Jun 17 2012

a[1] = a[2] = 0; a[n_] := a[n] = If[OddQ[n], a[(n - 1)/2], a[n/2] + Boole[Mod[n/2, 4] == 3]]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Oct 22 2012, after Ralf Stephan *)

a(n) = hammingweight(bitnegimply(bitand(n>>1, n>>2), n));
vector(102, i, a(i))  \\  Gheorghe Coserea, Sep 07 2015

a(2n) = a(n) + [n congruent to 3 mod 4], a(2n+1) = a(n). - Ralf Stephan, Aug 22 2003

a(n) = A213629(n,6) for n > 5. - Reinhard Zumkeller, Jun 17 2012

A056976 Number of blocks of {0, 1, 0} in the binary expansion of n.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein

Gheorghe Coserea, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digit Block

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

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, 1, 0}], {n, 1, 102} ] (* Jean-François Alcover, Oct 23 2012 *)
Table[SequenceCount[IntegerDigits[n,2],{0,1,0},Overlaps->True],{n,120}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 11 2019 *)

a(n) = {
  if (n < 10, return(0));
  my(k = logint(n,2) - 1);
  hammingweight(bitnegimply(n>>1, bitor(n, n >> 2))) - !bittest(n,k)
};
vector(102, i, a(i))  \\ Gheorghe Coserea, Sep 17 2015

a(2n) = a(n) + [n>1 and n congruent to 1 mod 4], a(2n+1) = a(n). - Ralf Stephan, Aug 22 2003

A056977 Number of blocks of {0, 1, 1} in binary expansion of n.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein

Gheorghe Coserea, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digit Block

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

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, 1, 1}], {n, 1, 102} ] (* Jean-François Alcover, Oct 23 2012 *)
Table[SequenceCount[IntegerDigits[n,2],{0,1,1}],{n,120}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 03 2019 *)

a(n) = {
  if (n < 11, return(0));
  my(k = logint(n,2) - 1);
  hammingweight(bitnegimply(bitand(n>>1, n), n>>2)) - bittest(n,k)
};
vector(102, i, a(i))  \\ Gheorghe Coserea, Sep 17 2015

a(2n) = a(n), a(2n+1) = a(n) + [n>1 and n congruent to 1 mod 4]. - Ralf Stephan, Aug 22 2003

A056975 Number of blocks of {0, 0, 1} in binary expansion of n.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein

Eric Weisstein's World of Mathematics, Digit Block

Cf. A007088 (binary expansion).

Other block counts: A014082, A056974, A056976, A056977, A056978, A056979, A056980.

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, 1}], {n, 1, 102} ] (* Jean-François Alcover, Oct 23 2012 *)
Table[SequenceCount[IntegerDigits[n,2],{0,0,1}],{n,110}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 26 2019 *)

a(2n) = a(n), a(2n+1) = a(n) + [n congruent to 0 mod 4]. - Ralf Stephan, Aug 22 2003

cp's OEIS Frontend

A033264 Number of blocks of {1,0} in the binary expansion of n.

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A037800 Number of occurrences of 01 in the binary expansion of n.

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

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A056978 Number of blocks of {1, 0, 0} in binary expansion of n.

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A056979 Number of blocks of {1, 0, 1} in binary expansion of n.

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A056980 Number of blocks of {1, 1, 0} in binary expansion of n.

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A056976 Number of blocks of {0, 1, 0} in the binary expansion of n.

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