A014081 -id:A014081 - cp's OEIS frontend

A020987 Zero-one version of Golay-Rudin-Shapiro sequence (or word).

0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0

Offset: 0

N. J. A. Sloane

This is (1-A020985(n))/2. See A020985, which is the main entry for this sequence, for more information. N. J. A. Sloane, Jun 06 2012

A word that is uniform primitive morphic, but not pure morphic. - N. J. A. Sloane, Jul 14 2018

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 78.
Dekking, Michel, Michel Mendes France, and Alf van der Poorten. "Folds." The Mathematical Intelligencer, 4.3 (1982): 130-138 & front cover, and 4:4 (1982): 173-181 (printed in two parts).
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
Lipshitz, Leonard, and A. van der Poorten. "Rational functions, diagonals, automata and arithmetic." In Number Theory, Richard A. Mollin, ed., Walter de Gruyter, Berlin (1990): 339-358.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jean-Paul Allouche, Schrödinger Operators with Rudin-Shapiro Potentials are not Palindromic, Journal of Mathematical Physics, volume 38, number 4, 1997, pages 1843-1848. And the author's copy. Section IV v_n = a(n) for the Rudin-Shapiro case given there i_0 = 0 and otherwise i_m = m+1 mod 2.
Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017
J. Brillhart and P. Morton, A case study in mathematical research: the Golay-Rudin-Shapiro sequence, Amer. Math. Monthly, 103 (1996) 854-869.
James D. Currie, Narad Rampersad, Kalle Saari, Luca Q. Zamboni, Extremal words in morphic subshifts, Discrete Math. 322 (2014), 53--60. MR3164037. See Sect. 8.
Michael Gilleland, Some Self-Similar Integer Sequences
A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic Schroedinger operators, Commun. Math. Phys. 174 (1995), 149-159.
L. Lipshitz and A. J. van der Poorten, Rational functions, diagonals, automata and arithmetic
H. Niederreiter and M. Vielhaber, Tree complexity and a doubly exponential gap between structured and random sequences, J. Complexity, 12 (1996), 187-198.
Aayush Rajasekaran, Narad Rampersad, Jeffrey Shallit, Overpals, Underlaps, and Underpals, In: Brlek S., Dolce F., Reutenauer C., Vandomme É. (eds) Combinatorics on Words, WORDS 2017, Lecture Notes in Computer Science, vol 10432.
Thomas Stoll, On digital blocks of polynomial values and extractions in the Rudin-Shapiro sequence, RAIRO - Theoretical Informatics and Applications (RAIRO: ITA), EDP Sciences, 2016, 50, pp. 93-99. .
Index entries for characteristic functions

Cf. A020985.
A014081(n) mod 2. Characteristic function of A022155.
Sequences mentioned in the Allouche et al. "Taxonomy" paper, listed by example number: 1: A003849, 2: A010060, 3: A010056, 4: A020985 and A020987, 5: A191818, 6: A316340 and A273129, 18: A316341, 19: A030302, 20: A063438, 21: A316342, 22: A316343, 23: A003849 minus its first term, 24: A316344, 25: A316345 and A316824, 26: A020985 and A020987, 27: A316825, 28: A159689, 29: A049320, 30: A003849, 31: A316826, 32: A316827, 33: A316828, 34: A316344, 35: A043529, 36: A316829, 37: A010060.

a020987 = (`div` 2) . (1 -) . a020985  -- Reinhard Zumkeller, Jun 06 2012

a[n_] := (1/2)*(1-(-1)^Count[Partition[IntegerDigits[n, 2], 2, 1], {1, 1}]); Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Dec 12 2014, after Robert G. Wilson v *)
(1 - RudinShapiro[Range[0, 100]])/2 (* Paolo Xausa, Jan 29 2025 *)

def A020987(n): return (n&(n>>1)).bit_count()&1 # Chai Wah Wu, Feb 11 2023

A245563 Table read by rows: row n gives list of lengths of runs of 1's in binary expansion of n, starting with low-order bits.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 2, 2, 3, 1, 3, 4, 5, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 1, 4, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 2, 2, 2, 3, 2, 3, 1, 3, 1, 3, 2, 3, 4, 1, 4, 5, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1

Offset: 0

N. J. A. Sloane, Aug 10 2014

A formula for A071053(n) depends on this table.

			Here are the run lengths for the numbers 0 through 21:
0, []
1, [1]
2, [1]
3, [2]
4, [1]
5, [1, 1]
6, [2]
7, [3]
8, [1]
9, [1, 1]
10, [1, 1]
11, [2, 1]
12, [2]
13, [1, 2]
14, [3]
15, [4]
16, [1]
17, [1, 1]
18, [1, 1]
19, [2, 1]
20, [1, 1]
21, [1, 1, 1]

Reinhard Zumkeller, Rows n = 0..1000 of triangle, flattened
Index entries for sequences related to binary expansion of n

Row sums = A000120 (the binary weight).
Cf. A245562, A071053.
Cf. A030308, A227736.
Row lengths are A069010.
The version for prime indices (instead of binary indices) is A124010.
Numbers with distinct run-lengths are A328592.
Numbers with equal run-lengths are A164707.
Cf. A003714, A014081, A328166, A328594, A328595, A328596.

import Data.List (group)
a245563 n k = a245563_tabf !! n !! k
a245563_row n = a245563_tabf !! n
a245563_tabf = [0] : map
   (map length . (filter ((== 1) . head)) . group) (tail a030308_tabf)
-- Reinhard Zumkeller, Aug 10 2014

for n from 0 to 128 do
lis:=[]; t1:=convert(n,base,2); L1:=nops(t1); out1:=1; c:=0;
for i from 1 to L1 do
if out1 = 1 and t1[i] = 1 then out1:=0; c:=c+1;
elif out1 = 0 and t1[i] = 1 then c:=c+1;
elif out1 = 1 and t1[i] = 0 then c:=c;
elif out1 = 0 and t1[i] = 0 then lis:=[op(lis),c]; out1:=1; c:=0;
fi;
if i = L1 and c>0 then lis:=[op(lis),c]; fi;
od:
lprint(n,lis);
od:

Join@@Table[Length/@Split[Join@@Position[Reverse[IntegerDigits[n,2]],1],#2==#1+1&],{n,0,100}] (* Gus Wiseman, Nov 03 2019 *)

from re import split
A245563_list = [0]
for n in range(1,100):
    A245563_list.extend(len(d) for d in split('0+',bin(n)[:1:-1]) if d != '')
# Chai Wah Wu, Sep 07 2014

A121016 Numbers whose binary expansion is properly periodic.

Original entry on oeis.org

3, 7, 10, 15, 31, 36, 42, 45, 54, 63, 127, 136, 153, 170, 187, 204, 221, 238, 255, 292, 365, 438, 511, 528, 561, 594, 627, 660, 682, 693, 726, 759, 792, 825, 858, 891, 924, 957, 990, 1023, 2047, 2080, 2145, 2184, 2210, 2275, 2340, 2405, 2457, 2470, 2535

Offset: 1

Jacob A. Siehler, Sep 08 2006

A finite sequence is aperiodic if its cyclic rotations are all different. - Gus Wiseman, Oct 31 2019

			For example, 204=(1100 1100)_2 and 292=(100 100 100)_2 belong to the sequence, but 30=(11110)_2 cannot be split into repeating periods.
From _Gus Wiseman_, Oct 31 2019: (Start)
The sequence of terms together with their binary expansions and binary indices begins:
   3:         11 ~ {1,2}
   7:        111 ~ {1,2,3}
   10:      1010 ~ {2,4}
   15:      1111 ~ {1,2,3,4}
   31:     11111 ~ {1,2,3,4,5}
   36:    100100 ~ {3,6}
   42:    101010 ~ {2,4,6}
   45:    101101 ~ {1,3,4,6}
   54:    110110 ~ {2,3,5,6}
   63:    111111 ~ {1,2,3,4,5,6}
  127:   1111111 ~ {1,2,3,4,5,6,7}
  136:  10001000 ~ {4,8}
  153:  10011001 ~ {1,4,5,8}
  170:  10101010 ~ {2,4,6,8}
  187:  10111011 ~ {1,2,4,5,6,8}
  204:  11001100 ~ {3,4,7,8}
  221:  11011101 ~ {1,3,4,5,7,8}
  238:  11101110 ~ {2,3,4,6,7,8}
  255:  11111111 ~ {1,2,3,4,5,6,7,8}
  292: 100100100 ~ {3,6,9}
(End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

A020330 is a subsequence.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose reversed binary expansion is Lyndon are A328596.
Numbers whose binary indices have equal run-lengths are A164707.
Cf. A000120, A003714, A014081, A065609, A069010, A275692, A328595.

PeriodicQ[n_, base_] := Block[{l = IntegerDigits[n, base]}, MemberQ[ RotateLeft[l, # ] & /@ Most@ Divisors@ Length@l, l]]; Select[ Range@2599, PeriodicQ[ #, 2] &]

is(n)=n=binary(n);fordiv(#n,d,for(i=1,#n/d-1, for(j=1,d, if(n[j]!=n[j+i*d], next(3)))); return(d<#n)) \\ Charles R Greathouse IV, Dec 10 2013

A034931 Triangle read by rows: Pascal's triangle (A007318) mod 4.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

The number of 3's in row n is given by 2^(A000120(n)-1) if A014081(n) is nonzero, else by 0 [Davis & Webb]. - R. J. Mathar, Jul 28 2017

			Triangle begins:
                 1
                1 1
               1 2 1
              1 3 3 1
             1 0 2 0 1
            1 1 2 2 1 1
           1 2 3 0 3 2 1
          1 3 1 3 3 1 3 1
         1 0 0 0 2 0 0 0 1
        1 1 0 0 2 2 0 0 1 1
       1 2 1 0 2 0 2 0 1 2 1
      1 3 3 1 2 2 2 2 1 3 3 1
  ...

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
Kenneth S. Davis and William A. Webb, Lucas' theorem for prime powers, European Journal of Combinatorics 11:3 (1990), pp. 229-233.
Kenneth S. Davis and William A. Webb, Pascal's triangle modulo 4, Fib. Quart., 29 (1991), 79-83.
Marc Evanstein, Hearing Pascal's Triangle Mod 4, YouTube video.
Ilya Gutkovskiy, Illustrations (triangle formed by reading Pascal's triangle mod m),.
James G. Huard, Blair K. Spearman, and Kenneth S. Williams, Pascal's triangle (mod 8), European Journal of Combinatorics 19:1 (1998), pp. 45-62.
Ivan Korec, Definability of Pascal's Triangles Modulo 4 and 6 and Some Other Binary Operations from Their Associated Equivalence Relations, Acta Univ. M. Belii Ser. Math. 4 (1996), pp. 53-66.
Index entries for triangles and arrays related to Pascal's triangle

Cf. A007318, A047999, A083093, A034930, A008975, A034932, A163000 (# 2's), A270438 (# 1's), A249732 (# 0's).
Cf. A000120, A014081.
Sequences based on the triangles formed by reading Pascal's triangle mod m: A047999 (m = 2), A083093 (m = 3), (this sequence) (m = 4), A095140 (m = 5), A095141 (m = 6), A095142 (m = 7), A034930(m = 8), A095143 (m = 9), A008975 (m = 10), A095144 (m = 11), A095145 (m = 12), A275198 (m = 14), A034932 (m = 16).

a034931 n k = a034931_tabl !! n !! k
a034931_row n = a034931_tabl !! n
a034931_tabl = iterate
   (\ws -> zipWith ((flip mod 4 .) . (+)) ([0] ++ ws) (ws ++ [0])) [1]
-- Reinhard Zumkeller, Mar 14 2015

A034931 := proc(n,k)
    modp(binomial(n,k),4) ;
end proc:
seq(seq(A034931(n,k),k=0..n),n=0..10); # R. J. Mathar, Jul 28 2017

Mod[ Flatten[ Table[ Binomial[n, k], {n, 0, 13}, {k, 0, n}]], 4] (* Robert G. Wilson v, May 26 2004 *)

C(n, k)=binomial(n, k)%4 \\ Charles R Greathouse IV, Aug 09 2016

f(n,k)=2*(bitand(n-k, k)==0);
T(n,j)=if(j==0,return(1)); my(k=logint(n,2),K=2^k,K1=K/2,L=n-K); if(LCharles R Greathouse IV, Aug 11 2016

from math import isqrt, comb
def A034931(n):
    g = (m:=isqrt(f:=n+1<<1))-(f<=m*(m+1))
    k = n-comb(g+1,2)
    if k.bit_count()+(g-k).bit_count()-g.bit_count()>1: return 0
    s, c, d = bin(g)[2:], 1, 0
    w = (bin(k)[2:]).zfill(l:=len(s))
    for i in range(0,l-1):
        r, t = s[i:i+2], w[i:i+2]
        if (x:=int(r,2)) < (y:=int(t,2)):
            d += (t[0]>r[0])+(t[1]>r[1])
        else:
            c = c*comb(x,y)&3
    d -= sum(1 for i in range(1,l-1) if w[i]>s[i])
    return (c<Chai Wah Wu, Jul 19 2025

T(n+1,k) = (T(n,k) + T(n,k-1)) mod 4. - Reinhard Zumkeller, Mar 14 2015

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

Original entry on oeis.org

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

A213629 In binary representation: T(n,k) = number of (possibly overlapping) occurrences of k in n, triangle read by rows, 1<=k<=n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jun 17 2012

The definition is based on the definition of pattern functions in the paper of Allouche and Shallit;
sum of n-th row = A029931(n);
T(n,1) = A000120(n);
T(n,2) = A033264(n) for n > 1;
T(n,3) = A014081(n) for n > 2;
T(n,4) = A056978(n) for n > 3;
T(n,5) = A056979(n) for n > 4;
T(n,6) = A056980(n) for n > 5;
T(n,7) = A014082(n) for n > 6;
T(n,k) = 0 for k with floor(n/2) < k < n;
T(n,n) = 1;
A122953(n) = Sum_{k=1..n} A057427(T(n,k));
A005811(n) = T(n,1) + T(n,2) - T(n,3);
A007302(n) = A000120(n) - sum (A213629(n,A136412(k))).

			The triangle begins:
.   1:                        1
.   2:                      1   1
.   3:                    2   0   1
.   4:                  1   1   0   1
.   5:                2   1   0   0   1
.   6:              2   1   1   0   0   1
.   7:            3   0   2   0   0   0   1
.   8:          1   1   0   1   0   0   0   1
.   9:        2   1   0   1   0   0   0   0   1
.  10:      2   2   0   0   1   0   0   0   0   1
.  11:    3   1   1   0   1   0   0   0   0   0   1
.  12:  2   1   1   1   0   1   0   0   0   0   0   1.

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
J.-P. Allouche, J. Shallit, The Ring of k-regular Sequences II, Example 4, p. 12
Index entries for sequences related to binary expansion of n

Cf. A030308, A007088.

import Data.List (inits, tails, isPrefixOf)
a213629 n k = a213629_tabl !! (n-1) !! (k-1)
a213629_row n = a213629_tabl !! (n-1)
a213629_tabl = map f $ tail $ inits $ tail $ map reverse a030308_tabf where
   f xss = map (\xs ->
           sum $ map (fromEnum . (xs `isPrefixOf`)) $ tails $ last xss) xss

t[n_, k_] := (idn = IntegerDigits[n, 2]; idk = IntegerDigits[k, 2]; ln = Length[idn]; lk = Length[idk]; For[cnt = 0; i = 1, i <= ln - lk + 1, i++, If[idn[[i ;; i + lk - 1]] == idk, cnt++]]; cnt); Table[t[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 22 2012 *)

A257250 Numbers n for which A256999(n) = n; numbers that cannot be made any larger by rotating (by one or more steps) the non-msb bits of their binary representation (with A080541 or A080542).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 12, 14, 15, 16, 24, 26, 28, 30, 31, 32, 48, 52, 56, 58, 60, 62, 63, 64, 96, 100, 104, 106, 112, 114, 116, 118, 120, 122, 124, 126, 127, 128, 192, 200, 208, 212, 224, 226, 228, 232, 234, 236, 240, 242, 244, 246, 248, 250, 252, 254, 255, 256, 384, 392, 400, 416, 420, 424, 426, 448, 450

Offset: 0

Antti Karttunen, May 16 2015

These correspond to the maximal (lexicographically largest) representatives selected from each equivalence class of binary necklaces. See the last example.
Indexing starts from zero, because a(0) = 0 is a special case.
If k is a member then so also is 2*k, i.e., k with 0 appended to the end of its binary representation.
If k is a member then so also is A004755(k), i.e., k with 1 prepended to the front of its binary representation.
One obtains A065609 if one erases the most significant bit of each term [as A053645(a(n))] and then discards any zero-terms produced from the terms that originally were powers of two (A000079).
First differs from A328607 in lacking 108, with binary expansion 1101100. If we define a dual-necklace to be a finite sequence that is lexicographically maximal (not minimal) among all of its cyclic rotations, these are numbers whose binary expansion, without the most significant digit, is a dual-necklace. - Gus Wiseman, Nov 04 2019

			For n = 5, with binary representation "101", if we rotate other bits than the most significant bit (that is, only the two rightmost digits "01") one step to either direction, we get "110" = 6 > 5, so 5 can be made larger by such rotations, and thus is NOT included in this sequence.
For n = 6, with binary representation "110", no such rotation will yield a larger number, and thus 6 is included in this sequence.
For n = 28, with binary representation "11100", if we rotate non-msb bits towards right, we get additional numbers 22, 19 and 25 (with binary representations "10110", "10011", "11001") before coming to 28 again, and 28 is the largest of these numbers, thus 28 is included in this sequence.
  Also, if we discard the most significant bit of each and consider them just as binary strings, then A053645(28) = 12 is the lexicographically largest representative of {"1100", "0110", "0011", "1001"}, which is the complete set of representatives for a particular equivalence class of binary necklaces, obtained by rotating all bits of binary string "1100" successively towards right or left.

Antti Karttunen, Table of n, a(n) for n = 0..16637
Wikipedia, Necklace (combinatorics)
Index entries for sequences related to necklaces

Complement: A257739.
Odd terms: A000225.
Subsequence of A065609.
Cf. A004755, A053645, A080541, A080542, A256999.
Subsequence: A258003.
The non-dual version is A328668.
The version involving all digits is A065609.
The non-dual reversed version is A328607.
Numbers whose reversed binary expansion is a necklace are A328595.
Binary necklaces are A000031.
Necklace compositions are A008965.
Cf. A000120, A001037, A003714, A014081, A069010, A275692, A328594, A328596.

reckQ[q_]:=Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
Select[Range[0,110],#<=1||reckQ[Rest[IntegerDigits[#,2]]]&] (* Gus Wiseman, Nov 04 2019 *)

A328593 Numbers whose binary indices have no consecutive divisible parts.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 12, 14, 16, 18, 20, 22, 24, 28, 30, 32, 40, 44, 46, 48, 50, 52, 54, 56, 60, 62, 64, 66, 68, 70, 72, 76, 78, 80, 82, 84, 86, 88, 92, 94, 96, 104, 108, 110, 112, 114, 116, 118, 120, 124, 126, 128, 132, 134, 144, 146, 148, 150, 152, 156, 158, 160

Offset: 1

Gus Wiseman, Oct 21 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The sequence of terms together with their binary expansions and binary indices begins:
   0:      0 ~ {}
   1:      1 ~ {1}
   2:     10 ~ {2}
   4:    100 ~ {3}
   6:    110 ~ {2,3}
   8:   1000 ~ {4}
  12:   1100 ~ {3,4}
  14:   1110 ~ {2,3,4}
  16:  10000 ~ {5}
  18:  10010 ~ {2,5}
  20:  10100 ~ {3,5}
  22:  10110 ~ {2,3,5}
  24:  11000 ~ {4,5}
  28:  11100 ~ {3,4,5}
  30:  11110 ~ {2,3,4,5}
  32: 100000 ~ {6}
  40: 101000 ~ {4,6}
  44: 101100 ~ {3,4,6}
  46: 101110 ~ {2,3,4,6}
  48: 110000 ~ {5,6}
  50: 110010 ~ {2,5,6}

The version for prime indices is A328603.
Numbers with no successive binary indices are A003714.
Partitions with no consecutive divisible parts are A328171.
Compositions without consecutive divisible parts are A328460.
Cf. A000120, A014081, A328187, A328508, A328594, A328595, A328598, A328600, A328608.

Select[Range[0,100],!MatchQ[Join@@Position[Reverse[IntegerDigits[#,2]],1],{_,x_,y_,_}/;Divisible[y,x]]&]

cp's OEIS Frontend

A020987 Zero-one version of Golay-Rudin-Shapiro sequence (or word).

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A245563 Table read by rows: row n gives list of lengths of runs of 1's in binary expansion of n, starting with low-order bits.

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A121016 Numbers whose binary expansion is properly periodic.

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A034931 Triangle read by rows: Pascal's triangle (A007318) mod 4.

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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.