A060140 Ordered set S defined by these rules: 0 and 1 are in S and if x is a nonzero number in S, then 3x and 9x+1 are in S.

0, 1, 3, 9, 10, 27, 28, 30, 81, 82, 84, 90, 91, 243, 244, 246, 252, 253, 270, 271, 273, 729, 730, 732, 738, 739, 756, 757, 759, 810, 811, 813, 819, 820, 2187, 2188, 2190, 2196, 2197, 2214, 2215, 2217, 2268, 2269, 2271, 2277, 2278, 2430, 2431, 2433, 2439

Offset: 0

Author

Clark Kimberling, Mar 05 2001

Keywords

• nonn

Comments

The numbers of the form 9x+1 occupy the same positions in S that 1 occupies in the infinite Fibonacci word (A003849).

These are Fibternary numbers: numbers whose ternary representations consist only of zeros and ones and do not have two consecutive ones. The sequence of Fibternary numbers can be constructed by writing out the Zeckendorf representations of nonnegative integers and then evaluating the result in ternary. These numbers are similar to Fibbinary numbers A003714, Tribbinary numbers A060140, and Tribternary numbers A356823. The number of Fibternary numbers less than any power of three is a Fibonacci number. We can generate Fibternary numbers recursively: Start by adding 0 and 1 to the sequence. Then, if x is a number in the sequence add 3x and 9x+1 to the sequence. Every nonnegative integer can be written as the sum of four Fibternary numbers. Every number has a Fibternary multiple. - Tanya Khovanova and PRIMES STEP Senior group, Aug 30 2022

Links

• Ivan Neretin, Table of n, a(n) for n = 0..10000
• Clark Kimberling, A Self-Generating Set and the Golden Mean, J. Integer Sequences, 3 (2000), Article 00.2.8.
• Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52 (2014), 195-202.

Crossrefs

Cf. A003714, A003849, A005836, A060138, A060142, A356823.

Programs

Mathematica

FromDigits[IntegerDigits[#, 2], 3] & /@ Select[Range[0, 165], BitAnd[#, 2*#] == 0 &] (* Amiram Eldar, Oct 28 2023 *)

Python

import heapq
from itertools import islice
def agen(): # generator of terms, using recursion in Comments
    yield 0; x, h = None, [1]
    while True:
        x = heapq.heappop(h)
        yield x
        for t in [3*x, 9*x+1]:  heapq.heappush(h, t)
print(list(islice(agen(), 51))) # Michael S. Branicky, Aug 30 2022

Python

def A060140(n):
    tlist, s = [1,2], 0
    while tlist[-1]+tlist[-2] <= n:
        tlist.append(tlist[-1]+tlist[-2])
    for d in tlist[::-1]:
        s <<= 1
        if d <= n:
            s += 1
            n -= d
    return int(bin(s)[2:],3) # Chai Wah Wu, May 22 2025

Formula

a(n) = A005836(A003714(n)+1). - Amiram Eldar, Oct 28 2023