A022341 a(n) = 4*A003714(n) + 1; the odd Fibbinary numbers.

1, 5, 9, 17, 21, 33, 37, 41, 65, 69, 73, 81, 85, 129, 133, 137, 145, 149, 161, 165, 169, 257, 261, 265, 273, 277, 289, 293, 297, 321, 325, 329, 337, 341, 513, 517, 521, 529, 533, 545, 549, 553, 577, 581, 585, 593, 597, 641, 645, 649, 657, 661, 673, 677, 681

Offset: 0

Author

Marc LeBrun

Keywords

• nonn
• easy

Comments

Numbers k such that (k+1) does not divide C(3k, k) - C(2k, k). - Benoit Cloitre, May 23 2004

Each term is the unique odd number a(n) = Sum_{i in S} 2^i such that n = Sum_{i in S} F_i, where F_i is the i-th Fibonacci number, A000045(i), and S is a set of nonnegative integers of which no two are adjacent. Note that this corresponds to adding F_0 to the Zeckendorf representation of n, which does not change the number being represented, because F_0 = 0. - Peter Munn, Sep 02 2022

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Estelle Basor, Brian Conrey, and Kent E. Morrison, Knots and ones, arXiv:1703.00990 [math.GT], 2017. See page 2.
• Linus Lindroos, Andrew Sills, and Hua Wang, Odd fibbinary numbers and the golden ration, Fib. Q., 52 (2014), 61-65.
• Linus Lindroos, Andrew Sills, and Hua Wang, Odd fibbinary numbers and the golden ration, Fib. Q., 52 (2014), 61-65.
• A. J. Macfarlane, On the fibbinary numbers and the Wythoff array, arXiv:2405.18128 [math.CO], 2024. See page 6.
• D. M. McKenna, Fibbinary Zippers in a Monoid of Toroidal Hamiltonian Cycles that Generate Hilbert-style Square-filling Curves, Enumerative Combinatorics and Applications, 2:2 #S2R13 (2021).

Crossrefs

Cf. A000045, A003714, A000846, A022340.

First column of A356875.

Programs

Maple

F:= combinat[fibonacci]:
b:= proc(n) local j;
      if n=0 then 0
    else for j from 2 while F(j+1)<=n do od;
         b(n-F(j))+2^(j-2)
      fi
    end:
a:= n-> 4*b(n)+1:
seq(a(n), n=0..70);  # Alois P. Heinz, May 15 2016

Mathematica

Select[Range[1, 511, 2], BitAnd[#, 2#] == 0 &] (* Alonso del Arte, Jun 18 2012 *)

Python

for n in range(1, 700, 2):
    if n*2 & n == 0:
        print(n, end=',')

Python

def A022341(n):
    tlist, s = [1,2], 0
    while tlist[-1]+tlist[-2] <= n: tlist.append(tlist[-1]+tlist[-2])
    for d in tlist[::-1]:
        if d <= n:
            s += 1
            n -= d
        s <<= 1
    return (s<<1)|1 # Chai Wah Wu, Apr 24 2025

Scala

(1 to 511 by 2).filter(n => (n & 2 * n) == 0) // Alonso del Arte, Apr 12 2020
(C#)
public static bool IsOddFibbinaryNum(this int n) => ((n & (n >> 1)) == 0) && (n % 2 == 1) ? true : false; // Frank Hollstein, Jul 07 2021

Extensions

More terms from Benoit Cloitre, May 23 2004 and Alonso del Arte, Jun 18 2012

Name edited by Peter Munn, Sep 02 2022