A092606 -id:A092606 - cp's OEIS frontend

A050292 a(2n) = 2n - a(n), a(2n+1) = 2n + 1 - a(n) (for n >= 0).

0, 1, 1, 2, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 20, 20, 21, 21, 22, 22, 23, 24, 25, 25, 26, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 36, 36, 37, 37, 38, 38, 39, 40, 41, 41, 42, 43, 44, 44, 45, 46, 47, 47, 48, 48, 49, 49, 50, 51, 52, 52, 53, 54

Offset: 0

Eric W. Weisstein

Note that the first equation implies a(0)=0, so there is no need to specify an initial value.
Maximal cardinality of a double-free subset of {1, 2, ..., n}, or in other words, maximal size of a subset S of {1, 2, ..., n} with the property that if x is in S then 2x is not. a(0)=0 by convention.
Least k such that a(k)=n is equal to A003159(n).
To construct the sequence: let [a, b, c, a, a, a, b, c, a, b, c, ...] be the fixed point of the morphism a -> abc, b ->a, c -> a, starting from a(1) = a, then write the indices of a, b, c, that of a being written twice; see A092606. - Philippe Deléham, Apr 13 2004
Number of integers from {1,...,n} for which the subtraction of 1 changes the parity of the number of 1's in their binary expansion. - Vladimir Shevelev, Apr 15 2010
Number of integers from {1,...,n} the factorization of which over different terms of A050376 does not contain 2. - Vladimir Shevelev, Apr 16 2010
a(n) modulo 2 is the Prouhet-Thue-Morse sequence A010060. Each number n appears A026465(n+1) times. - Philippe Deléham, Oct 19 2011
Another way of stating the last two comments from Philippe Deléham: the sequence can be obtained by replacing each term of the Thue-Morse sequence A010060 by the run number that term is in. - N. J. A. Sloane, Dec 31 2013

			Examples for n = 1 through 8: {1}, {1}, {1,3}, {1,3,4}, {1,3,4,5}, {1,3,4,5}, {1,3,4,5,7}, {1,3,4,5,7}.
Binary expansion of 5 is 101, so Sum{i>=0} b_i*(-1)^i = 2. Therefore a(5) = 10/3 + 2/3 = 4. - _Vladimir Shevelev_, Apr 15 2010

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.26.
Wang, E. T. H. "On Double-Free Sets of Integers." Ars Combin. 28, 97-100, 1989.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Steven R. Finch, Triple-Free Sets of Integers [From Steven Finch, Apr 20 2019]
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Eric Weisstein's World of Mathematics, Double-Free Set.

Bisection of A123087.

Cf. A001045, A050291-A050296, A050321, A035263, A121701, A030308, A080277, A120385, A197911.

a050292 n = a050292_list !! (n-1)
a050292_list = scanl (+) 0 a035263_list
-- Reinhard Zumkeller, Jan 21 2013

A050292:=n->add((-1)^k*floor(n/2^k), k=0..n); seq(A050292(n), n=0..100); # Wesley Ivan Hurt, Feb 14 2014

a[n_] := a[n] = If[n < 2, 1, n - a[Floor[n/2]]]; Table[ a[n], {n, 1, 75}]
Join[{0},Accumulate[Nest[Flatten[#/.{0->{1,1},1->{1,0}}]&,{0},7]]] (* Harvey P. Dale, Apr 29 2018 *)

PARI
```
a(n)=if(n<2,1,n-a(floor(n/2)))
```

from sympy.ntheory import digits
def A050292(n): return ((n<<1)+sum((0,1,-1,0)[i] for i in digits(n,4)[1:]))//3 # Chai Wah Wu, Jan 30 2025

Partial sums of A035263. Close to (2/3)*n.
a(n) = A123087(2*n) = n - A123087(n). - Max Alekseyev, Mar 05 2023
From Benoit Cloitre, Nov 24 2002: (Start)
a(1)=1, a(n) = n - a(floor(n/2));
a(n) = (2/3)*n + (1/3)*A065359(n);
more generally, for m>=0, a(2^m*n) - 2^m*a(n) = A001045(m)*A065359(n) where A001045(m) = (2^m - (-1)^m)/3 is the Jacobsthal sequence;
a(A039004(n)) = (2/3)*A039004(n);
a(2*A039004(n)) = 2*a(A039004(n));
a(A003159(n)) = n;
a(A003159(n)-1) = n-1;
a(n) mod 2 = A010060(n) the Thue-Morse sequence;
a(n+1) - a(n) = A035263(n+1);
a(n+2) - a(n) = abs(A029884(n)).
(End)
G.f.: (1/(x-1)) * Sum_{i>=0} (-1)^i*x^(2^i)/(x^(2^i)-1). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 17 2003
a(n) = Sum_{k>=0} (-1)^k*floor(n/2^k). - Benoit Cloitre, Jun 03 2003
a(A091785(n)) = 2n; a(A091855(n)) = 2n-1. - Philippe Deléham, Mar 26 2004
a(2^n) = (2^(n+1) + (-1)^n)/3. - Vladimir Shevelev, Apr 15 2010
If n = Sum_{i>=0} b_i*2^i is the binary expansion of n, then a(n) = 2n/3 + (1/3)Sum_{i>=0} b_i*(-1)^i. Thus a(n) = 2n/3 + O(log(n)). - Vladimir Shevelev, Apr 15 2010
Moreover, the equation a(3m)=2m has infinitely many solutions, e.g., a(3*2^k)=2*2^k; on the other hand, a((4^k-1)/3)=(2*(4^k-1))/9+k/3, i.e., limsup |a(n)-2n/3| = infinity. - Vladimir Shevelev, Feb 23 2011
a(n) = Sum_{k>=0} A030308(n,k)*A001045(k+1). - Philippe Deléham, Oct 19 2011
From Peter Bala, Feb 02 2013: (Start)
Product_{n >= 1} (1 + x^((2^n - (-1)^n)/3 )) = (1 + x)^2(1 + x^3)(1 + x^5)(1 + x^11)(1 + x^21)... = 1 + sum {n >= 1} x^a(n) = 1 + 2x + x^2 + x^3 + 2x^4 + 2x^5 + .... Hence this sequence lists the numbers representable as a sum of distinct Jacobsthal numbers A001045 = [1, 1', 3, 5, 11, 21, ...], where we distinguish between the two occurrences of 1 by writing them as 1 and 1'. For example, 9 occurs twice in the present sequence because 9 = 5 + 3 + 1 and 9 = 5 + 3 + 1'. Cf. A197911 and A080277. See also A120385.
(End)

Extended with formula by Christian G. Bower, Sep 15 1999
Corrected and extended by Reinhard Zumkeller, Aug 16 2006
Extended with formula by Philippe Deléham, Oct 19 2011
Entry revised to give a simpler definition by N. J. A. Sloane, Jan 03 2014

A003158 A self-generating sequence (see Comments in A003156 for the definition).

Original entry on oeis.org

2, 7, 10, 13, 18, 23, 28, 31, 34, 39, 42, 45, 50, 53, 56, 61, 66, 71, 74, 77, 82, 87, 92, 95, 98, 103, 108, 113, 116, 119, 124, 127, 130, 135, 138, 141, 146, 151, 156, 159

Offset: 1

N. J. A. Sloane

nonn

Numbers not of the form Sum_{i>=2} e_i*A001045(i), with e(i) = 0 or 1.

Indices of b in the sequence closed under a -> abc, b -> a, c -> a, starting with a(1) = a; see A092606 where a = 0, b = 2, c = 1. - Philippe Deléham, Apr 12 2004

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972. See p. 67.
L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., Representations for a special sequence, Fibonacci Quarterly 10.5 (1972), 499-518, 550.

Cf. A001045, A003156, A003157, A079523, A092606.

def A003158(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, s = n+x, bin(x)[2:]
        l = len(s)
        for i in range(l&1,l,2):
            c -= int(s[i])+int('0'+s[:i],2)
        return c
    return bisection(f,n,n)+n-1 # Chai Wah Wu, Jan 29 2025

a(n) = A003157(n) - 1 = A079523(n) + n. - Philippe Deléham, Feb 22 2004

Definition clarified by N. J. A. Sloane, Dec 26 2020

A003156 A self-generating sequence (see Comments for definition).

Original entry on oeis.org

1, 4, 5, 6, 9, 12, 15, 16, 17, 20, 21, 22, 25, 26, 27, 30, 33, 36, 37, 38, 41, 44, 47, 48, 49, 52, 55, 58, 59, 60, 63, 64, 65, 68, 69, 70, 73, 76, 79, 80, 81, 84, 85, 86, 89, 90, 91, 94, 97, 100, 101, 102, 105, 106, 107, 110, 111, 112, 115, 118, 121, 122, 123, 126, 129, 132

Offset: 1

N. J. A. Sloane

nonn

From N. J. A. Sloane, Dec 26 2020: (Start)
The best definitions of the triple [this sequence, A003157, A003158] are as the rows a(n), b(n), c(n) of the table:
n: 1, 2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, ...
a: 1, 4,  5,  6,  9, 12, 15, 16, 17, 20, 21, 22, ...
b: 3, 8, 11, 14, 19, 24, 29, 32, 35, 40, 43, 46, ...
c: 2, 7, 10, 13, 18, 23, 28, 31, 34, 39, 42, 45, ...
where a(1)=1, b(1)=3, c(1)=2, and thereafter
a(n) = mex{a(i), b(i), c(i), ib(n) = a(n) + 2*n,
c(n) = b(n) - 1.
Then a,b,c form a partition of the positive integers.
Note that there is another triple of sequences (A003144, A003145, A003146) also called a, b, c and also a partition of the positive integers, in a different paper by the same authors (Carlitz-Scovelle-Hoggatt) in the same volume of the same journal.
(End)
a(n) is the number of ones before the n-th zero in the Feigenbaum sequence A035263. - Philippe Deléham, Mar 27 2004
Number of odd numbers before the n-th even number in A007413, A007913, A001511, A029883, A033485, A035263, A036585, A065882, A065883, A088172, A092412. - Philippe Deléham, Apr 03 2004
Indices of a in the sequence closed under a -> abc, b -> a, c -> a, starting with a(1) = a; see A092606 where a = 0, b = 2, c = 1. - Philippe Deléham, Apr 12 2004

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Jean-Paul Allouche, On the morphism 1 -> 121, 2 -> 12221, Preprint, 2024 [Local copy, with permission]. See also Optimization, Discrete Mathematics and Applications to Data Sciences, Springer Optimization and Its Applications, Springer, Cham, 2025, and author's site, CNRS France, 2024. See p. 6.
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Fibonacci representations of higher order, Fib. Quart., 10 (1972), 43-69. [Here a, b, c are A003144, A003145, A003146.]
L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., Representations for a special sequence, Fibonacci Quarterly 10.5 (1972), 499-518, 550. [A003156, A003157, A003158 are on page 500.]
Robbert Fokkink and Gandhar Joshi, On Cloitre's hiccup sequences, arXiv:2507.16956 [math.CO], 2025. See pp. 8, 17

Cf. A001511, A003157, A003158, A003159, A007413, A007913, A029883, A033485, A035263, A036554, A036585, A056196, A065882, A065883, A072939, A079523, A072939, A080426, A088172, A092412, A092606.

A003157 A self-generating sequence (see Comments in A003156 for the definition).

Original entry on oeis.org

3, 8, 11, 14, 19, 24, 29, 32, 35, 40, 43, 46, 51, 54, 57, 62, 67, 72, 75, 78, 83, 88, 93, 96, 99, 104, 109, 114, 117, 120, 125, 128, 131, 136, 139, 142, 147, 152, 157, 160

Offset: 1

N. J. A. Sloane

nonn

Indices of c in the sequence closed under a -> abc, b -> a, c -> a, starting with a(1) = a; see A092606 where a = 0, b = 2, c = 1. - Philippe Deléham, Apr 12 2004

These are the positions of 1 in A286044; complement of A286045; conjecture: a(n)/n -> 4. - Clark Kimberling, May 07 2017

			As a word, A286044 = 001000010010010000100..., in which 1 is in positions a(n) for n>=1.  - _Clark Kimberling_, May 07 2017

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Clark Kimberling, Table of n, a(n) for n = 1..10000
L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., Representations for a special sequence, Fibonacci Quarterly 10.5 (1972), 499-518, 550.

Cf. A003156, A003158, A010060, A286044, A286045.

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 9] (* Thue-Morse, A010060 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"011" -> "0"}]
st = ToCharacterCode[w1] - 48 (* A286044 *)
Flatten[Position[st, 0]]  (* A286045 *)
Flatten[Position[st, 1]]  (* A003157 *)
(* Clark Kimberling, May 07 2017 *)

def A003157(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, s = n+x, bin(x)[2:]
        l = len(s)
        for i in range(l&1,l,2):
            c -= int(s[i])+int('0'+s[:i],2)
        return c
    return bisection(f,n,n)+n # Chai Wah Wu, Jan 29 2025

Numbers n such that A003159(n) is even. a(n) = A003158(n) + 1 = A036554(n) + n. - Philippe Deléham, Feb 22 2004

A339584 A ternary sequence: a(n) = 1 if n is in A003156, 2 if n is in A003157, 3 if n is in A003158.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Dec 26 2020

nonn

This has the same relationship to A003156-A003158 as A092782 does to A003144-A003146.

Cf. A003156, A003157, A003158; A003144, A003145, A003146, A080843, A092606, A092782.

Nest[Flatten[# /. {1 -> {1, 3, 2}, 2 -> 1, 3 -> 1}] &, {1}, 8] (* John Keith, Dec 28 2020 after Robert G. Wilson v's program for A092782. *)

a(n) = A092606(n) + 1. - John Keith, Dec 28 2020

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cp's OEIS Frontend

A050292 a(2n) = 2n - a(n), a(2n+1) = 2n + 1 - a(n) (for n >= 0).

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A003158 A self-generating sequence (see Comments in A003156 for the definition).

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A003156 A self-generating sequence (see Comments for definition).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Python

Formula

Extensions

A003157 A self-generating sequence (see Comments in A003156 for the definition).

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Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Python

Formula

A339584 A ternary sequence: a(n) = 1 if n is in A003156, 2 if n is in A003157, 3 if n is in A003158.

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