A356823 -id:A356823 - cp's OEIS frontend

A003726 Numbers with no 3 adjacent 1's in binary expansion.

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 48, 49, 50, 51, 52, 53, 54, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 80, 81, 82

Offset: 1

N. J. A. Sloane

Positions of zeros in A014082. Could be called "tribbinary numbers" by analogy with A003714. - John Keith, Mar 07 2022

The sequence of Tribbinary numbers can be constructed by writing out the Tribonacci representations of nonnegative integers and then evaluating the result in binary. These numbers are similar to Fibbinary numbers A003714, Fibternary numbers A003726, and Tribternary numbers A356823. The number of Tribbinary numbers less than any power of two is a Tribonacci number. We can generate Tribbinary numbers recursively: Start by adding 0 and 1 to the sequence. Then, if x is a number in the sequence add 2x, 4x+1, and 8x+3 to the sequence. The n-th Tribbinary number is even if the n-th term of the Tribonacci word is a. Respectively, the n-th Tribbinary number is of the form 4x+1 if the n-th term of the Tribonacci word is b, and the n-th Tribbinary number is of the form 8x+3 if the n-th term of the Tribonacci word is c. Every nonnegative integer can be written as the sum of two Tribbinary numbers. Every number has a Tribbinary multiple. - Tanya Khovanova and PRIMES STEP Senior, Aug 30 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Index entries for 2-automatic sequences.

Cf. A278038 (binary), A063037, A000073, A014082 (number of 111).
Cf. A004781 (complement).
Cf. A007088; A003796 (no 000), A004745 (no 001), A004746 (no 010), A004744 (no 011), A003754 (no 100), A004742 (no 101), A004743 (no 110).

a003726 n = a003726_list !! (n - 1)
a003726_list = filter f [0..] where
   f x = x < 7 || (x `mod` 8) < 7 && f (x `div` 2)
-- Reinhard Zumkeller, Jun 03 2012

Select[Range[0, 82], SequenceCount[IntegerDigits[#, 2], {1, 1, 1}] == 0 &] (* Michael De Vlieger, Dec 23 2019 *)

is(n)=!bitand(bitand(n, n<<1), n<<2) \\ Charles R Greathouse IV, Feb 11 2017

There are A000073(n+3) terms of this sequence with at most n bits. In particular, a(A000073(n+3)+1) = 2^n. - Charles R Greathouse IV, Oct 22 2021

Sum_{n>=2} 1/a(n) = 9.516857810319139410424631558212354346868048230248717360943194590798113163384... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 13 2022

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.

Original entry on oeis.org

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

Clark Kimberling, Mar 05 2001

nonn

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

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.

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

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

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

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

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

cp's OEIS Frontend

A003726 Numbers with no 3 adjacent 1's in binary expansion.

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