A014081 -id:A014081 - cp's OEIS frontend

A245195 a(n) = 2^A014081(n).

1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 2, 2, 4, 8, 1, 1, 1, 2, 1, 1, 2, 4, 2, 2, 2, 4, 4, 4, 8, 16, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 2, 2, 4, 8, 2, 2, 2, 4, 2, 2, 4, 8, 4, 4, 4, 8, 8, 8, 16, 32, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 2, 2, 4, 8, 1, 1, 1, 2, 1, 1, 2, 4, 2, 2, 2, 4, 4, 4, 8, 16, 2, 2, 2, 4, 2

Offset: 0

N. J. A. Sloane, Jul 24 2014

This sequence provides a bridge between A245180 (and, presumably, A160239) and A014081.
See A245196 for more about this class of sequences.
Run length transform of A011782: 1,1,2,4,8,16,32,64,... - Chai Wah Wu, Oct 19 2016

Chai Wah Wu and Robert Israel, Table of n, a(n) for n = 0..10000
Robert Israel, Proof that A277560 is the same as A245195 [This will be modified to reflect the fact that the two sequences have now been merged]
Chai Wah Wu, Sums of products of binomial coefficients mod 2 and run length transforms of sequences, arXiv:1610.06166 [math.CO], 2016.

Cf. A014081, A245180, A160239, A048896, A245196, A005725, A005940, A011782, A106737, A162510.

# This Maple program applies more generally to a sequence where the recurrence across a block is as follows. The parameters to be set are the sequence G(0), G(1), G(2), ... (the final terms in the blocks), and the multiplier m.
# For n in the range 2^(k-1) <= n < 2^k, write n = 2^k-2^r+j, with 0 <= r <= k-1 and 0 <= j < 2^(r-1), and j=0 if r=0. Then
# (if j=0) a(2^k-2^r) = G(k-r-1),
# (if j>0) a(2^k-2^r+j) = m*G(k-r-1)*a(j).
# Since Maple gives its lists an offset of 1, it is necessary to add 1 to the arguments of G.
# For the present sequence, G(n)=2^n and m=1.
G:=[seq(2^n,n=0..30)];
m:=1;
f:=proc(n) option remember; global m,G; local k,r,j,np;
if n <= 2 then G[0+1] elif n=3 then G[1+1]
elif n=4 then G[0+1] elif n=5 then m*G[0+1] elif n=6 then G[1+1] elif n=7 then G[2+1]
else
   k:=1+floor(log[2](n)); np:=2^k-n;
   if np=1 then r:=0; j:=0; else r:=1+floor(log[2](np-1)); j:=2^r-np; fi;
   if j=0 then G[k-r-1+1]; else m*G[k-r-1+1]*f(j); fi;
fi;
end;
[seq(f(n),n=1..520)]:
# Setting G(n) = A083424(n) and m = 8 gives A245180. Setting G(n) = 2^n and m = 2 gives A048896.
A245195:=n->add(binomial(n,2*k)*binomial(n,k) mod 2, k=0..floor(n/2)): seq(A245195(n), n=0..200); # Wesley Ivan Hurt, Nov 01 2016

Table[Sum[Mod[Binomial[n, 2 k] Binomial[n, k], 2], {k, 0, n}], {n, 0, 85}] (* Michael De Vlieger, Oct 21 2016 *)

a(n) = 2^hammingweight(bitand(n, n>>1)) \\ Charles R Greathouse IV, Jul 16 2016

a(n) = sum(k=0, n, binomial(n, 2*k)*binomial(n,k) % 2); \\ Michel Marcus, Oct 21 2016

from _future_ import division
def A277560(n):
    return sum(int(not (~n & 2*k) | (~n & k)) for k in range(n//2+1))

def A245195(n): return 1<<(n&(n>>1)).bit_count() # Chai Wah Wu, Feb 11 2023

The entries may be arranged into blocks of sizes 1,2,4,8,...:
B_0: 1,
B_1: 1, 2,
B_2: 1, 1, 2, 4,
B_3: 1, 1, 1, 2, 2, 2, 4, 8,
B_4: 1, 1, 1, 2, 1, 1, 2, 4, 2, 2, 2, 4, 4, 4, 8, 16,
B_5: 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 2, 2, 4, 8, 2, 2, 2, 4, 2, 2, 4, 8, 4, 4, 4, 8, 8, 8, 16, 32,
...
Consider the block B_{k-1} containing terms a(2^(k-1)), a(2^(k-1)+1), ..., a(2^k-1). It is convenient to index the terms working backwards from the next, 2^k-th, term. For n in the range 2^(k-1) <= n < 2^k, write n = 2^k-2^r+j, with 0 <= r <= k-1 and 0 <= j < 2^(r-1), and j=0 if r=0. Then
(if j=0) a(2^k-2^r) = 2^(k-r-1),
(if j>0) a(2^k-2^r+j) = 2^(k-r-1)*a(j).
a(n) = A162510(A005940(1+n)). - Antti Karttunen, Oct 29 2016
From Robert Israel, Nov 02 2016: (Start)
a(2*k)   = a(k).
a(4*k+1) = a(k).
a(4*k+3) = 2*a(2*k+1).
G.f. g(x) satisfies g(x) = x + (2*x+1)*g(x^2) - x*g(x^4). (End)
Also, a(n) = Sum_{k=0..floor(n/2)} ((binomial(n,2k)*binomial(n,k)) mod 2). - Chai Wah Wu, Oct 19 2016 and Robert Israel, Nov 04 2016. For proof, see the article by Chai Wah Wu, Sums of products of binomial coefficients mod 2 and run length transforms of sequences, arXiv:1610.06166, or the Robert Israel link.

Changed offset to 0, merged former entry A277560 from Chai Wah Wu (Oct 19 2016) with this sequence. - N. J. A. Sloane, Nov 05 2016

A346422 a(n) = (1 + A014081(n))*a(A053645(n)) for n > 0 with a(0) = 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 4, 2, 2, 6, 24, 1, 1, 1, 4, 1, 1, 4, 18, 2, 2, 2, 12, 6, 6, 24, 120, 1, 1, 1, 4, 1, 1, 4, 18, 1, 1, 1, 8, 4, 4, 18, 96, 2, 2, 2, 12, 2, 2, 12, 72, 6, 6, 6, 48, 24, 24, 120, 720, 1, 1, 1, 4, 1, 1, 4, 18, 1, 1, 1, 8, 4, 4, 18, 96

Offset: 0

Mikhail Kurkov, Aug 08 2021 [verification needed]

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^16.

Cf. A003714 (positions of 1's), A014081, A036987, A053645.

Nest[Append[#1, (1 + Count[Partition[IntegerDigits[#2, 2], 2, 1], {1, 1}]) #1[[#2 - 2^Floor@ Log2[#2] + 1]]] & @@ {#, Length[#]} &, {1}, 79] (* Michael De Vlieger, Feb 04 2022 *)

a(n)=if(n==0, 1, (1+b(n))*a(c(n)))
b(n)=if(n==1, 0, if(n%4<2, b(n\4), b(n\2) + n%2)) \\ A014081
c(n)=if(n==1, 0, 2*c(n\2) + n%2) \\ A053645

a(n) = my(f=1,ret=1); if(n, for(i=0,logint(n,2), if(bittest(n,i), ret*=(f+=bittest(n,i-1))))); ret; \\ Kevin Ryde, Aug 25 2021

from functools import lru_cache
from re import split
@lru_cache(maxsize=None)
def A346422(n): return 1 if n <= 1 else A346422(int((s:= bin(n)[2:])[1:],2))*(1+sum(len(d)-1 for d in split('0+', s) if d != '')) # Chai Wah Wu, Feb 04 2022

a(n) = (1 + A014081(n))*a(A053645(n)) for n > 0 with a(0) = 1.
a(4n+1) = a(2n) = a(n) for n > 0 with a(0) = a(1) = 1.
a(4n+3) = 2!*b(n), a(8n+11) = 2*2!*b'(n), b'(2n) = b(n), b'(2n+1) = b'(n),
a(8n+7) = 3!*c(n), a(16n+23) = 3*3!*c'(n), c'(2n) = c(n), c'(2n+1) = c'(n),
a(16n+15) = 4!*d(n), a(32n+47) = 4*4!*d'(n), d'(2n) = d(n), d'(2n+1) = d'(n),
a(32n+31) = 5!*e(n), a(64n+95) = 5*5!*e'(n), e'(2n) = e(n), e'(2n+1) = e'(n),
and so on, i.e.,
a(2^m*(n+1) - 1) = m!*z(n), a(2^m*(2n+3) - 1) = m*m!*z'(n), z'(2n) = z(n), z'(2n+1) = z'(n).
From that, we get:
a(2^3*(2n+1) + 3) = 2*a(4n+3), a(2^3*(2n+1) + 11) = a(8n+11),
a(2^4*(2n+1) + 7) = 3*a(8n+7), a(2^4*(2n+1) + 23) = a(16n+23),
a(2^5*(2n+1) + 15) = 4*a(16n+15), a(2^5*(2n+1) + 47) = a(32n+47),
a(2^6*(2n+1) + 31) = 5*a(32n+31), a(2^6*(2n+1) + 95) = a(64n+95),
and so on, i.e.,
a(2^m*(4n+3) - 1) = m*a(2^m*(n+1) - 1), a(2^m*(4n+5) - 1) = a(2^m*(2n+3) - 1).
Let
p(n) = 0 if A036987(n) = 1 otherwise p(2n+1) = 2 + p(n), p(2n) = 2 - (n mod 2) for n > 0 with p(0) = p(1) = 0,
q(2n+1) = 2^(n+2) - 1, q(2n) = 2^(n+2) + q(2n-1) for n > 0 with q(1) = 3,
p_1(n) = 0 if A036987(n) = 1 otherwise q(p(n)) for n > 0 with p_1(0) = 0,
p_2(n) = 0 if A036987(n) = 1 otherwise p_2(2n+1) = p_2(n), p_2(2n) = floor((n - 1)/2) for n > 0 with p_2(0) = p_2(1) = 0,
so
a(4n+3) = (log_2(4n+4))! if A036987(4n+3) = 1 otherwise (1 + (p(n) mod 2)*(p(n) + 1)/2)*a(p_2(n)*2^(floor(p(n)/2) + 2) + p_1(n)) for n >= 0.
a((4^n - 1)/3) = 1 for n >= 0.
a(2^m*(2^n - 1)) = n! for n > 0, m >= 0.

A136545 Semi-chaotic binary digit sum/product sequence modeled on a Rudin-Shapiro-type sequence like A014081.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 5, 3, 6, 4, 7, 4, 7, 4, 11, 3, 12, 4, 13, 5, 14, 5, 15, 4, 17, 5, 17, 5, 17, 5, 22, 3, 23, 4, 24, 5, 25, 5, 27, 5, 29, 6, 29, 6, 29, 6, 32, 4, 33

Offset: 1

Roger L. Bagula, Mar 24 2008

Cf. A014081.

Clear[s, k, n] k[n_] := Apply[Plus, Table[1 - Mod[n - Floor[n/2^m], 2]Mod[n - Floor[n/2^(m - 1)], 2], {m, 1, Floor[(n)*Log[2]]}]]; a = Table[k[n], {n, 1, 50}]

a(n)=Sum[1 - Mod[n - Floor[n/2^m], 2]Mod[n - Floor[n/2^(m - 1)], 2],{m, 1, Floor[(n)*Log[2]]}]

A136546 Three-part semi-chaotic binary digit sum/product sequence modeled on a Rudin-Shapiro-type sequence like A014081.

Original entry on oeis.org

0, 0, 1, 1, 1, 3, 3, 4, 4, 4, 5, 7, 7, 8, 9, 10, 9, 10, 11, 11, 11, 13, 13, 15, 15, 16, 16, 18, 18, 19, 20, 21, 20, 21, 22, 22, 22, 24, 25, 25, 25, 26, 26, 28, 28, 29, 30, 32, 31, 32

Offset: 1

Roger L. Bagula, Mar 24 2008

Cf. A014081.

Clear[s, k, n] k[n_] := Apply[ Plus, Table[1 - Mod[n - Floor[n/2^m], 2] +Mod[n - Floor[n/2^m], 2]Mod[n - Floor[n/2^(m - 1)], 2], {m, 1, Floor[(n)*Log[2]]}]]; a = Table[k[n], {n, 1, 50}]

a(n)=Sum[1 - Mod[n - Floor[n/2^m], 2] + Mod[n - Floor[n/2^m], 2]Mod[n - Floor[n/2^(m - 1)], 2],{m, 1, Floor[(n)*Log[2]]}]

A003714 Fibbinary numbers: if n = F(i1) + F(i2) + ... + F(ik) is the Zeckendorf representation of n (i.e., write n in Fibonacci number system) then a(n) = 2^(i1 - 2) + 2^(i2 - 2) + ... + 2^(ik - 2). Also numbers whose binary representation contains no two adjacent 1's.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 10, 16, 17, 18, 20, 21, 32, 33, 34, 36, 37, 40, 41, 42, 64, 65, 66, 68, 69, 72, 73, 74, 80, 81, 82, 84, 85, 128, 129, 130, 132, 133, 136, 137, 138, 144, 145, 146, 148, 149, 160, 161, 162, 164, 165, 168, 169, 170, 256, 257, 258, 260, 261, 264

Offset: 0

N. J. A. Sloane

The name "Fibbinary" is due to Marc LeBrun.
"... integers whose binary representation contains no consecutive ones and noticed that the number of such numbers with n bits was fibonacci(n)". [posting to sci.math by Bob Jenkins (bob_jenkins(AT)burtleburtle.net), Jul 17 2002]
From Benoit Cloitre, Mar 08 2003: (Start)
A number m is in the sequence if and only if C(3m, m) (or equally, C(3m, 2m)) is odd.
a(n) == A003849(n) (mod 2). (End)
Numbers m such that m XOR 2*m = 3*m. - Reinhard Zumkeller, May 03 2005. [This implies that A003188(2*a(n)) = 3*a(n) holds for all n.]
Numbers whose base-2 representation contains no two adjacent ones. For example, m = 17 = 10001_2 belongs to the sequence, but m = 19 = 10011_2 does not. - Ctibor O. Zizka, May 13 2008
m is in the sequence if and only if the central Stirling number of the second kind S(2*m, m) = A007820(m) is odd. - O-Yeat Chan (math(AT)oyeat.com), Sep 03 2009
A000120(3*a(n)) = 2*A000120(a(n)); A002450 is a subsequence.
Every nonnegative integer can be expressed as the sum of two terms of this sequence. - Franklin T. Adams-Watters, Jun 11 2011
Subsequence of A213526. - Arkadiusz Wesolowski, Jun 20 2012
This is also the union of A215024 and A215025 - see the Comment in A014417. - N. J. A. Sloane, Aug 10 2012
The binary representation of each term m contains no two adjacent 1's, so we have (m XOR 2m XOR 3m) = 0, and thus a two-player Nim game with three heaps of (m, 2m, 3m) stones is a losing configuration for the first player. - V. Raman, Sep 17 2012
Positions of zeros in A014081. - John Keith, Mar 07 2022
These numbers are similar to Fibternary numbers A003726, Tribbinary numbers A060140 and Tribternary numbers. This sequence is a subsequence of Fibternary numbers A003726. The number of Fibbinary numbers less than any power of two is a Fibonacci number. We can generate this sequence recursively: start with 0 and 1; then, if x is in the sequence add 2x and 4x+1 to the sequence. The Fibbinary numbers have the property that the n-th Fibbinary number is even if the n-th term of the Fibonacci word is a. Respectively, the n-th Fibbinary number is odd (of the form 4x+1) if the n-th term of the Fibonacci word is b. Every number has a Fibbinary multiple. - Tanya Khovanova and PRIMES STEP Senior, Aug 30 2022
This is the ordered set S of numbers defined recursively by: 0 is in S; if x is in S, then 2*x and 4*x + 1 are in S. See Kimberling (2006) Example 3, in references below. - Harry Richman, Jan 31 2024

			From _Joerg Arndt_, Jun 11 2011: (Start)
In the following, dots are used for zeros in the binary representation:
  a(n)  binary(a(n))  n
    0:    .......     0
    1:    ......1     1
    2:    .....1.     2
    4:    ....1..     3
    5:    ....1.1     4
    8:    ...1...     5
    9:    ...1..1     6
   10:    ...1.1.     7
   16:    ..1....     8
   17:    ..1...1     9
   18:    ..1..1.    10
   20:    ..1.1..    11
   21:    ..1.1.1    12
   32:    .1.....    13
   33:    .1....1    14
   34:    .1...1.    15
   36:    .1..1..    16
   37:    .1..1.1    17
   40:    .1.1...    18
   41:    .1.1..1    19
   42:    .1.1.1.    20
   64:    1......    21
   65:    1.....1    22
(End)

Donald E. Knuth, The Art of Computer Programming: Fundamental Algorithms, Vol. 1, 2nd ed., Addison-Wesley, 1973, pp. 85, 493.

G. C. Greubel and T. D. Noe, Table of n, a(n) for n = 0..5000 (terms 0 to 1363 by T. D. Noe)
J.-P. Allouche, J. Shallit, and G. Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math., Vol. 292, No. 1-3 (2005), pp. 1-15.
Joerg Arndt, Matters Computational (The Fxtbook), pp. 74-77, pp. 754-756.
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Marc Chamberland and Karl Dilcher, A Binomial Sum Related to Wolstenholme's Theorem, J. Number Theory, Vol. 171, Issue 11 (Nov. 2009), pp. 2659-2672. See Lemma 4.2 p. 2668.
O-Yeat Chan and Dante Manna, Divisibility properties of Stirling numbers of the second kind.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
David Eppstein, Generating fibbinary numbers, three ways, 2021.
Clark Kimberling, Affinely recursive sets and orderings of languages, Discrete Math., Vol. 274, No. 1-3 (2004), pp. 147-160.
Roman S. Klyujkov, Quick Fibbinary Numbers Addition, C++ function with test program.
Ron Knott, Rabbit Sequence in Zeckendorf Expansion (A003714).
Linus Lindroos, Andrew Sills, and Hua Wang, Odd fibbinary numbers and the golden ratio, Fib. Q., Vol. 52, No. 1 (2014), pp. 61-65; alternative link.
A. J. Macfarlane, On the fibbinary numbers and the Wythoff array, arXiv:2405.18128 [math.CO], 2024. See pages 1, 5.
Douglas M. McKenna, Fibbinary Zippers in a Monoid of Toroidal Hamiltonian Cycles that Generate Hilbert-Style Square-Filling Curves, ECA 2:2 (2022) Article S2R13.
Joris Nieuwveld, Fractions, Functions and Folding. A Novel Link between Continued Fractions, Mahler Functions and Paper Folding, Master's Thesis, arXiv:2108.11382 [math.NT], 2021.
Yaron Shany and Amit Berman, The Generating Idempotent Is a Minimum-Weight Codeword for Some Binary BCH Codes, arXiv:2408.08218 [cs.IT], 2024. See p. 10.
N. J. A. Sloane, Table of n, a(n) (base 10), a(n) (base 2), for n = 0..1000.
Chai Wah Wu, Record values in appending and prepending bitstrings to runs of binary digits, arXiv:1810.02293 [math.NT], 2018.
Index entries for 2-automatic sequences.
Index entries for sequences defined by congruent products between domains N and GF(2)[X].
Index entries for sequences defined by congruent products under XOR.

A007088(a(n)) = A014417(n) (same sequence in binary). Complement: A004780. Char. function: A085357. Even terms: A022340, odd terms: A022341. First difference: A129761.
Other sequences based on similar restrictions on binary expansion: A003726 & A278038, A003754, A048715, A048718, A107907, A107909.
Cf. A000045, A005203, A005590, A007895, A037011, A048728, A048679, A056017, A060112, A072649, A083368, A089939, A106027, A106028, A116361.
3*a(n) is in A001969.
Cf. also A215022-A215025, A242407.
Cf. A014081 (count 11 bits).

import Data.Set (Set, singleton, insert, deleteFindMin)
a003714 n = a003714_list !! n
a003714_list = 0 : f (singleton 1) where
   f :: Set Integer -> [Integer]
   f s = m : (f $ insert (4*m + 1) $ insert (2*m) s')
         where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 03 2012, Feb 07 2012

A003714 := proc(n)
    option remember;
    if n < 3 then
        n ;
    else
        2^(A072649(n)-1) + procname(n-combinat[fibonacci](1+A072649(n))) ;
    end if;
end proc:
seq(A003714(n),n=0..10) ;
# To produce a table giving n, a(n) (base 10), a(n) (base 2) - from N. J. A. Sloane, Sep 30 2018
# binary: binary representation of n, in human order
binary:=proc(n) local t1,L;
if n<0 then ERROR("n must be nonnegative"); fi;
if n=0 then return([0]); fi;
t1:=convert(n,base,2); L:=nops(t1);
[seq(t1[L+1-i],i=1..L)];
end;
for n from 0 to 100 do t1:=A003714(n); lprint(n, t1, binary(t1)); od:

fibBin[n_Integer] := Block[{k = Ceiling[Log[GoldenRatio, n Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; FromDigits[fr, 2]]; Table[fibBin[n], {n, 0, 61}] (* Robert G. Wilson v, Sep 18 2004 *)
Select[Range[0, 270], ! MemberQ[Partition[IntegerDigits[#, 2], 2, 1], {1, 1}] &] (* Harvey P. Dale, Jul 17 2011 *)
Select[Range[256], BitAnd[#, 2 #] == 0 &] (* Alonso del Arte, Jun 18 2012 *)
With[{r = Range[10^5]}, Pick[r, BitAnd[r, 2 r], 0]] (* Eric W. Weisstein, Aug 18 2017 *)
Select[Range[0, 299], SequenceCount[IntegerDigits[#, 2], {1, 1}] == 0 &] (* Requires Mathematica version 10 or later. -- Harvey P. Dale, Dec 06 2018 *)

msb(n)=my(k=1); while(k<=n, k<<=1); k>>1
for(n=1,1e4,k=bitand(n,n<<1);if(k,n=bitor(n,msb(k)-1),print1(n", "))) \\ Charles R Greathouse IV, Jun 15 2011

select( is_A003714(n)=!bitand(n,n>>1), [0..266])
{(next_A003714(n,t)=while(t=bitand(n+=1,n<<1), n=bitor(n,1<A003714(t)) \\ M. F. Hasler, Nov 30 2021

for n in range(300):
    if 2*n & n == 0:
        print(n, end=",") # Alex Ratushnyak, Jun 21 2012

def A003714(n):
    tlist, s = [1,2], 0
    while tlist[-1]+tlist[-2] <= n:
        tlist.append(tlist[-1]+tlist[-2])
    for d in tlist[::-1]:
        s *= 2
        if d <= n:
            s += 1
            n -= d
    return s # Chai Wah Wu, Jun 14 2018

def fibbinary():
    x = 0
    while True:
        yield x
        y = ~(x >> 1)
        x = (x - y) & y # Falk Hüffner, Oct 23 2021
(C++)
/* start with x=0, then repeatedly call x=next_fibrep(x): */
ulong next_fibrep(ulong x)
{
    // 2 examples:         //  ex. 1             //  ex.2
    //                     // x == [*]0 010101   // x == [*]0 01010
    ulong y = x | (x>>1);  // y == [*]? 011111   // y == [*]? 01111
    ulong z = y + 1;       // z == [*]? 100000   // z == [*]? 10000
    z = z & -z;            // z == [0]0 100000   // z == [0]0 10000
    x ^= z;                // x == [*]0 110101   // x == [*]0 11010
    x &= ~(z-1);           // x == [*]0 100000   // x == [*]0 10000
    return x;
}
/* Joerg Arndt, Jun 22 2012 */

(0 to 255).filter(n => (n & 2 * n) == 0) // Alonso del Arte, Apr 12 2020
(C#)
public static bool IsFibbinaryNum(this int n) => ((n & (n >> 1)) == 0) ? true : false; // Frank Hollstein, Jul 07 2021

No two adjacent 1's in binary expansion.
Let f(x) := Sum_{n >= 0} x^Fibbinary(n). (This is the generating function of the characteristic function of this sequence.) Then f satisfies the functional equation f(x) = x*f(x^4) + f(x^2).
a(0) = 0, a(1) = 1, a(2) = 2, a(n) = 2^(A072649(n) - 1) + a(n - A000045(1 + A072649(n))). - Antti Karttunen
It appears that this sequence gives m such that A082759(3*m) is odd; or, probably equivalently, m such that A037011(3*m) = 1. - Benoit Cloitre, Jun 20 2003
If m is in the sequence then so are 2*m and 4*m + 1. - Henry Bottomley, Jan 11 2005
A116361(a(n)) <= 1. - Reinhard Zumkeller, Feb 04 2006
A085357(a(n)) = 1; A179821(a(n)) = a(n). - Reinhard Zumkeller, Jul 31 2010
a(n)/n^k is bounded (but does not tend to a limit), where k = 1.44... = A104287. - Charles R Greathouse IV, Sep 19 2012
a(n) = a(A193564(n+1))*2^(A003849(n) + 1) + A003849(n) for n > 0. - Daniel Starodubtsev, Aug 05 2021
There are Fibonacci(n+1) terms with up to n bits in this sequence. - Charles R Greathouse IV, Oct 22 2021
Sum_{n>=1} 1/a(n) = 3.704711752910469457886531055976801955909489488376627037756627135425780134020... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 12 2022

Edited by Antti Karttunen, Feb 21 2006
Cross reference to A007820 added (into O-Y.C. comment) by Jason Kimberley, Sep 14 2009
Typo corrected by Jeffrey Shallit, Sep 26 2014

A069010 Number of runs of 1's in the binary representation of n.

Original entry on oeis.org

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, 1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 2, 2, 1, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 2, 2, 2, 3, 3, 3, 3, 4, 3, 3, 2, 3, 3, 3, 2, 3

Offset: 0

Henry Bottomley, Apr 02 2002

a(n) is also the number of distinct parts in the integer partition having viabin number n. The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [2,2,2,1]. The southeast border of its Ferrers board yields 10100, leading to the viabin number 20. - Emeric Deutsch, Jul 24 2017

Positions of first occurrences of k are A002450(k). - John Keith, Aug 30 2021

			a(11) = 2 since 11 is 1011 in binary with two runs of 1's.
a(12) = 1 since 12 is 1100 in binary with one run of 1's.

Tilman Piesk (terms 0..9999) & Antti Karttunen, Table of n, a(n) for n = 0..16384
Louis Marin, Counting Polyominoes in a Rectangle b X h, arXiv:2406.16413 [cs.DM], 2024. See p. 148.
Vladimir Shevelev, Two analogs of Thue-Morse sequence, arXiv:1603.04434 [math.NT], 2016.
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

Cf. A000035, A000120, A005811, A014081, A033264, A037800.
Cf. A268411 (parity of the terms), A268412 (positions of even terms), A268415 (of odd terms).
Cf. A002450 (positions of record highs).
Cf. also A227349, A246588.

f:= proc(n) option remember; if n::even then procname(n/2)
elif n mod 4 = 1 then 1 + procname((n-1)/2) else  procname((n-1)/2) fi end proc:
f(0):= 0:
map(f, [$0..1000]); # Robert Israel, Sep 06 2015

Count[Split@ IntegerDigits[#, 2], n_ /; First@ n == 1] & /@ Range[0, 120] (* Michael De Vlieger, Sep 05 2015 *)

a(n) = (1 + (hammingweight(bitxor(n, n>>1)))) >> 1;  \\ Gheorghe Coserea, Sep 05 2015

def A069010(n):
    return sum(1 for d in bin(n)[2:].split('0') if len(d)) # Chai Wah Wu, Nov 04 2016

(define (A069010 n) (/ (+ (A005811 n) (A000035 n)) 2)) ;; Antti Karttunen, Feb 05 2016

a(n) = ceiling(A005811(n)/2) = A005811(n) - A033264(n). If 2^k <= n < 3*2^(k-1) then a(n) = a(n-2^k)+1; if 3*2^(k-1) <= n < 2^(k+1) then a(n) = a(n-2^k).
a(2n) = a(n), a(2n+1) = a(n) + [n is even]. - Ralf Stephan, Aug 20 2003
G.f.: (1/(1-x)) * Sum_{k>=0} (t/(1+t))/(1+t^2), where t=x^2^k. - Ralf Stephan, Sep 07 2003
a(n) = A000120(n) - A014081(n) = A037800(n) + 1, n>0. - Ralf Stephan, Sep 10 2003

A328594 Numbers whose binary expansion is aperiodic.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Gus Wiseman, Oct 22 2019

nonn

A finite sequence is aperiodic if all of its cyclic rotations are distinct. See A000740 or A027375 for details.

Also numbers k such that the k-th composition in standard order is aperiodic. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. - Gus Wiseman, Apr 28 2020

			The sequence of terms together with their binary expansions and binary indices begins:
   0:     0 ~ {}
   1:     1 ~ {1}
   2:    10 ~ {2}
   4:   100 ~ {3}
   5:   101 ~ {1,3}
   6:   110 ~ {2,3}
   8:  1000 ~ {4}
   9:  1001 ~ {1,4}
  11:  1011 ~ {1,2,4}
  12:  1100 ~ {3,4}
  13:  1101 ~ {1,3,4}
  14:  1110 ~ {2,3,4}
  16: 10000 ~ {5}
  17: 10001 ~ {1,5}
  18: 10010 ~ {2,5}
  19: 10011 ~ {1,2,5}
  20: 10100 ~ {3,5}
  21: 10101 ~ {1,3,5}
  22: 10110 ~ {2,3,5}
  23: 10111 ~ {1,2,3,5}
  24: 11000 ~ {4,5}

The complement is A121016.
The version for prime indices is A085971.
Numbers without proper integer roots are A007916.
Necklaces are A328595.
Lyndon words are A328596.
Aperiodic compositions are A000740.
Aperiodic binary sequences are A027375.
Cf. A000120, A000939, A014081, A065609, A069010, A275692, A323867, A334030.

aperQ[q_]:=Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
Select[Range[0,100],aperQ[IntegerDigits[#,2]]&]

A020985 The Rudin-Shapiro or Golay-Rudin-Shapiro sequence (coefficients of the Shapiro polynomials).

Original entry on oeis.org

1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, -1, -1, -1, 1, -1, 1, 1, 1, -1, 1, 1, -1, 1, -1, -1, -1, 1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, -1, -1, -1, 1, -1, -1, -1, -1, 1, -1, -1, 1, -1, 1, 1, 1, -1, -1, -1, 1, -1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, -1, -1, -1, 1, -1, 1

Offset: 0

N. J. A. Sloane

Other names are the Rudin-Shapiro or Golay-Rudin-Shapiro infinite word.
The Shapiro polynomials are defined by P_0 = Q_0 = 1; for n>=0, P_{n+1} = P_n + x^(2^n)*Q_n, Q_{n+1} = P_n - x^(2^n)*Q_n. Then P_n = Sum_{m=0..2^n-1} a(m)*x^m, where the a(m) (the present sequence) do not depend on n. - N. J. A. Sloane, Aug 12 2016
Related to paper-folding sequences - see the Mendès France and Tenenbaum article.
a(A022155(n)) = -1; a(A203463(n)) = 1. - Reinhard Zumkeller, Jan 02 2012
a(n) = 1 if and only if the numbers of 1's and runs of 1's in binary representation of n have the same parity: A010060(n) = A268411(n); otherwise, when A010060(n) = 1 - A268411(n), a(n) = -1. - Vladimir Shevelev, Feb 10 2016. Typo corrected and comment edited by Antti Karttunen, Jul 11 2017
A word that is uniform primitive morphic, but not pure morphic. - N. J. A. Sloane, Jul 14 2018
Named after the Austrian-American mathematician Walter Rudin (1921-2010), the mathematician Harold S. Shapiro (1928-2021) and the Swiss mathematician and physicist Marcel Jules Edouard Golay (1902-1989). - Amiram Eldar, Jun 13 2021

Jean-Paul Allouche and Jeffrey Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 78 and many other pages.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jean-Paul Allouche, Lecture notes on automatic sequences, Krakow October 2013.
Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, and Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017
Jean-Paul Allouche and M. Mendes France, Automata and Automatic Sequences, in: Axel F. and Gratias D. (eds), Beyond Quasicrystals. Centre de Physique des Houches, Vol. 3, Springer, Berlin, Heidelberg, pp. 293-367, 1995; DOI https://doi.org/10.1007/978-3-662-03130-8_11.
Jean-Paul Allouche and M. Mendes France, Automata and Automatic Sequences, in: Axel F. and Gratias D. (eds), Beyond Quasicrystals. Centre de Physique des Houches, vol 3. Springer, Berlin, Heidelberg, pp. 293-367, 1995; DOI https://doi.org/10.1007/978-3-662-03130-8_11. [Local copy]
Jean-Paul Allouche and Jonathan Sondow, Summation of rational series twisted by strongly B-multiplicative coefficients, arXiv:1408.5770 [math.NT], 2014; Electron. J. Combin., 22 #1 (2015) P1.59; see pp.9-10.
Joerg Arndt, Matters Computational (The Fxtbook), section 1.16.5 "The Golay-Rudin-Shapiro sequence", pp.44-45
Scott Balchin and Dan Rust, Computations for Symbolic Substitutions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.1.
John Brillhart and L. Carlitz, Note on the Shapiro polynomials, Proc. Amer. Math. Soc., Vol. 25 (1970), pp. 114-118.
John Brillhart and Patrick Morton, Über Summen von Rudin-Shapiroschen Koeffizienten, (German) Illinois J. Math., Vol. 22, No. 1 (1978), pp. 126-148. MR0476686 (57 #16245). - From _N. J. A. Sloane_, Jun 06 2012
John Brillhart and Patrick Morton, A case study in mathematical research: the Golay-Rudin-Shapiro sequence, Amer. Math. Monthly, Vol. 103 (1996) pp. 854-869.
James D. Currie, Narad Rampersad, Kalle Saari, and Luca Q. Zamboni, Extremal words in morphic subshifts, arXiv:1301.4972 [math.CO], 2013.
James D. Currie, Narad Rampersad, Kalle Saari, and Luca Q. Zamboni, Extremal words in morphic subshifts, Discrete Math., Vol. 322 (2014), pp. 53-60. MR3164037. See Sect. 8.
Michel Dekking, Michel Mendes France and Alf van der Poorten, Folds, The Mathematical Intelligencer, Vol. 4, No. 3 (1982), pp. 130-138.
Michel Dekking, Michel Mendes France and Alf van der Poorten, Folds II. Symmetry disturbed, The Mathematical Intelligencer, Vol. 4, No. 4 (1982), pp. 173-181.
Arturas Dubickas, Heights of squares of Littlewood polynomials and infinite series, Ann. Polon. Math., Vol. 105 (2012), pp. 145-163. - From _N. J. A. Sloane_, Dec 16 2012
Russell Jay Hendel, A Family of Sequences Generalizing the Thue Morse and Rudin Shapiro Sequences, arXiv:2505.20547 [cs.FL], 2025. See p. 2.
Albertus Hof, Oliver Knill and Barry Simon, Singular continuous spectrum for palindromic Schroedinger operators, Commun. Math. Phys., Vol. 174, No. 1 (1995), pp. 149-159.
Philip Lafrance, Narad Rampersad and Randy Yee, Some properties of a Rudin-Shapiro-like sequence, arXiv:1408.2277 [math.CO], 2014.
D. H. Lehmer and Emma Lehmer, Picturesque exponential sums. II, Journal für die reine und angewandte Mathematik, Vol. 318 (1980), pp. 1-19.
Michel Mendès France and Gérald Tenenbaum, Dimension des courbes planes, papiers pliés et suites de Rudin-Shapiro. (French) Bull. Soc. Math. France, Vol. 109, No. 2 (1981), pp. 207-215. MR0623789 (82k:10073).
Luke Schaeffer and Jeffrey Shallit, Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences, Electronic Journal of Combinatorics, Vol. 23, No. 1 (2016), #P1.25.
Harold S. Shapiro, Extremal problems for polynomials and power series, Ph.D. Diss. Massachusetts Institute of Technology, 1952.
Vladimir Shevelev, Two analogs of Thue-Morse sequence, arXiv:1603.04434 [math.NT], 2016.
Eric Weisstein's World of Mathematics, Rudin-Shapiro Sequence.

Cf. A022155, A005943 (factor complexity), A014081.
Cf. A020987 (0-1 version), A020986 (partial sums), A203531 (run lengths), A033999, A380667 (first differences).
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.

a020985 n = a020985_list !! n
a020985_list = 1 : 1 : f (tail a020985_list) (-1) where
   f (x:xs) w = x : x*w : f xs (0 - w)
-- Reinhard Zumkeller, Jan 02 2012

A020985 := proc(n) option remember; if n = 0 then 1 elif n mod 2 = 0 then A020985(n/2) else (-1)^((n-1)/2 )*A020985( (n-1)/2 ); fi; end;

a[0] = 1; a[1] = 1; a[n_?EvenQ] := a[n] = a[n/2]; a[n_?OddQ] := a[n] = (-1)^((n-1)/2)* a[(n-1)/2]; a /@ Range[0, 80] (* Jean-François Alcover, Jul 05 2011 *)
a[n_] := 1 - 2 Mod[Length[FixedPointList[BitAnd[#, # - 1] &, BitAnd[n, Quotient[n, 2]]]], 2] (* Jan Mangaldan, Jul 23 2015 *)
Array[RudinShapiro, 81, 0] (* JungHwan Min, Dec 22 2016 *)

A020985(n)=(-1)^A014081(n)  \\ M. F. Hasler, Jun 06 2012

def a014081(n): return sum([((n>>i)&3==3) for i in range(len(bin(n)[2:]) - 1)])
def a(n): return (-1)**a014081(n) # Indranil Ghosh, Jun 03 2017

def A020985(n): return -1 if (n&(n>>1)).bit_count()&1 else 1 # Chai Wah Wu, Feb 11 2023

a(0) = a(1) = 1; thereafter, a(2n) = a(n), a(2n+1) = a(n) * (-1)^n. [Brillhart and Carlitz, in proof of theorem 4]
a(0) = a(1) = 1, a(2n) = a(n), a(2n+1) = a(n)*(1-2*(n AND 1)), where AND is the bitwise AND operator. - Alex Ratushnyak, May 13 2012
Brillhart and Morton (1978) list many properties.
a(n) = (-1)^A014081(n)  = (-1)^A020987(n) = 1-2*A020987(n). - M. F. Hasler, Jun 06 2012
Sum_{n >= 1} a(n-1)*(8*n^2+4*n+1)/(2*n*(2*n+1)*(4*n+1)) = 1; see Allouche and Sondow, 2015. - Jean-Paul Allouche and Jonathan Sondow, Mar 21 2015

A328592 Numbers whose binary expansion has all different lengths of runs of 1's.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 11, 12, 13, 14, 15, 16, 19, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 35, 38, 39, 44, 46, 47, 48, 49, 50, 52, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 67, 70, 71, 76, 78, 79, 88, 92, 94, 95, 96, 97, 98, 100, 103, 104, 110, 111, 112, 113, 114

Offset: 1

Gus Wiseman, Oct 20 2019

nonn

Also numbers whose binary indices have different lengths of runs of successive parts. 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 complement is {5, 9, 10, 17, 18, 20, 21, 27, ...}.

			The sequence of terms together with their binary expansions and binary indices begins:
   0:     0 ~ {}
   1:     1 ~ {1}
   2:    10 ~ {2}
   3:    11 ~ {1,2}
   4:   100 ~ {3}
   6:   110 ~ {2,3}
   7:   111 ~ {1,2,3}
   8:  1000 ~ {4}
  11:  1011 ~ {1,2,4}
  12:  1100 ~ {3,4}
  13:  1101 ~ {1,3,4}
  14:  1110 ~ {2,3,4}
  15:  1111 ~ {1,2,3,4}
  16: 10000 ~ {5}
  19: 10011 ~ {1,2,5}
  22: 10110 ~ {2,3,5}
  23: 10111 ~ {1,2,3,5}
  24: 11000 ~ {4,5}
  25: 11001 ~ {1,4,5}
  26: 11010 ~ {2,4,5}

The version for prime indices is A130091.
The binary expansion of n has A069010(n) runs of 1's.
The lengths of runs of 1's in the binary expansion of n are row n of A245563.
Numbers whose binary expansion has equal lengths of runs of 1's are A164707.
Cf. A000120, A003714, A014081, A098859, A121016, A328593, A328595, A328596.

Select[Range[0,100],UnsameQ@@Length/@Split[Join@@Position[Reverse[IntegerDigits[#,2]],1],#2==#1+1&]&]

A164707 A positive integer n is included if all runs of 1's in binary n are of the same length.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 17, 18, 20, 21, 24, 27, 28, 30, 31, 32, 33, 34, 36, 37, 40, 41, 42, 48, 51, 54, 56, 60, 62, 63, 64, 65, 66, 68, 69, 72, 73, 74, 80, 81, 82, 84, 85, 96, 99, 102, 108, 112, 119, 120, 124, 126, 127, 128, 129, 130, 132, 133, 136

Offset: 1

Leroy Quet, Aug 23 2009

Clarification: A binary number consists of "runs" completely of 1's alternating with runs completely of 0's. No two or more runs all of the same digit are adjacent.

This sequence contains in part positive integers that each contain one run of 1's. For those members of this sequence each with at least two runs of 1's, see A164709.

			From _Gus Wiseman_, Oct 31 2019: (Start)
The sequence of terms together with their binary expansions and binary indices begins:
   1:      1 ~ {1}
   2:     10 ~ {2}
   3:     11 ~ {1,2}
   4:    100 ~ {3}
   5:    101 ~ {1,3}
   6:    110 ~ {2,3}
   7:    111 ~ {1,2,3}
   8:   1000 ~ {4}
   9:   1001 ~ {1,4}
  10:   1010 ~ {2,4}
  12:   1100 ~ {3,4}
  14:   1110 ~ {2,3,4}
  15:   1111 ~ {1,2,3,4}
  16:  10000 ~ {5}
  17:  10001 ~ {1,5}
  18:  10010 ~ {2,5}
  20:  10100 ~ {3,5}
  21:  10101 ~ {1,3,5}
  24:  11000 ~ {4,5}
  27:  11011 ~ {1,2,4,5}
(End)

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A164708, A164709, A164710.
The version for prime indices is A072774.
The binary expansion of n has A069010(n) runs of 1's.
Numbers whose runs are all of different lengths are A328592.
Partitions with equal multiplicities are A047966.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose reversed binary expansion is a necklace are A328595.
Numbers whose reversed binary expansion is a Lyndon word are A328596.
Cf. A000120, A003714, A014081, A065609, A070939, A121016, A245563, A275692.

isA164707 := proc(n) local bdg,arl,lset ; bdg := convert(n,base,2) ; lset := {} ; arl := -1 ; for p from 1 to nops(bdg) do if op(p,bdg) = 1 then if p = 1 then arl := 1 ; else arl := arl+1 ; end if; else if arl > 0 then lset := lset union {arl} ; end if; arl := 0 ; end if; end do ; if arl > 0 then lset := lset union {arl} ; end if; return (nops(lset) <= 1 ); end proc: for n from 1 to 300 do if isA164707(n) then printf("%d,",n) ; end if; end do; # R. J. Mathar, Feb 27 2010

Select[Range@ 140, SameQ @@ Map[Length, Select[Split@ IntegerDigits[#, 2], First@ # == 1 &]] &] (* Michael De Vlieger, Aug 20 2017 *)

foreach(1..140){
    %runs=();
    $runs{$}++ foreach split /0+/, sprintf("%b",$);
    print "$_, " if 1==keys(%runs);
}
# Ivan Neretin, Nov 09 2015

Extended beyond 42 by R. J. Mathar, Feb 27 2010

cp's OEIS Frontend

A245195 a(n) = 2^A014081(n).

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A346422 a(n) = (1 + A014081(n))*a(A053645(n)) for n > 0 with a(0) = 1.

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A136545 Semi-chaotic binary digit sum/product sequence modeled on a Rudin-Shapiro-type sequence like A014081.

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A136546 Three-part semi-chaotic binary digit sum/product sequence modeled on a Rudin-Shapiro-type sequence like A014081.

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A003714 Fibbinary numbers: if n = F(i1) + F(i2) + ... + F(ik) is the Zeckendorf representation of n (i.e., write n in Fibonacci number system) then a(n) = 2^(i1 - 2) + 2^(i2 - 2) + ... + 2^(ik - 2). Also numbers whose binary representation contains no two adjacent 1's.

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A069010 Number of runs of 1's in the binary representation of n.

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