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

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

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

Examples

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

References

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

Links

• 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.

Crossrefs

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

Programs

Mathematica

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 *)

Python

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

Formula

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