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

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

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

References

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

Links

• 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.

Crossrefs

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

Programs

Python

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

Formula

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

Extensions

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