cp's OEIS Frontend

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

%I A022341 #86 Apr 24 2025 17:23:45
%S A022341 1,5,9,17,21,33,37,41,65,69,73,81,85,129,133,137,145,149,161,165,169,
%T A022341 257,261,265,273,277,289,293,297,321,325,329,337,341,513,517,521,529,
%U A022341 533,545,549,553,577,581,585,593,597,641,645,649,657,661,673,677,681
%N A022341 a(n) = 4*A003714(n) + 1; the odd Fibbinary numbers.
%C A022341 Numbers k such that (k+1) does not divide C(3k, k) - C(2k, k). - _Benoit Cloitre_, May 23 2004
%C A022341 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
%H A022341 Alois P. Heinz, <a href="/A022341/b022341.txt">Table of n, a(n) for n = 0..10000</a>
%H A022341 Estelle Basor, Brian Conrey, and Kent E. Morrison, <a href="https://arxiv.org/abs/1703.00990">Knots and ones</a>, arXiv:1703.00990 [math.GT], 2017. See page 2.
%H A022341 Linus Lindroos, Andrew Sills, and Hua Wang, <a href="http://home.dimacs.rutgers.edu/~asills/Fibbinary/Fibbinary.pdf">Odd fibbinary numbers and the golden ration</a>, Fib. Q., 52 (2014), 61-65.
%H A022341 Linus Lindroos, Andrew Sills, and Hua Wang, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/52-1/LindroosSillsWang.pdf">Odd fibbinary numbers and the golden ration</a>, Fib. Q., 52 (2014), 61-65.
%H A022341 A. J. Macfarlane, <a href="https://arxiv.org/abs/2405.18128">On the fibbinary numbers and the Wythoff array</a>, arXiv:2405.18128 [math.CO], 2024. See page 6.
%H A022341 D. M. McKenna, <a href="http://ecajournal.haifa.ac.il/Volume2022/ECA2022_S2A13.pdf">Fibbinary Zippers in a Monoid of Toroidal Hamiltonian Cycles that Generate Hilbert-style Square-filling Curves</a>, Enumerative Combinatorics and Applications, 2:2 #S2R13 (2021).
%p A022341 F:= combinat[fibonacci]:
%p A022341 b:= proc(n) local j;
%p A022341       if n=0 then 0
%p A022341     else for j from 2 while F(j+1)<=n do od;
%p A022341          b(n-F(j))+2^(j-2)
%p A022341       fi
%p A022341     end:
%p A022341 a:= n-> 4*b(n)+1:
%p A022341 seq(a(n), n=0..70);  # _Alois P. Heinz_, May 15 2016
%t A022341 Select[Range[1, 511, 2], BitAnd[#, 2#] == 0 &] (* _Alonso del Arte_, Jun 18 2012 *)
%o A022341 (Python)
%o A022341 for n in range(1, 700, 2):
%o A022341     if n*2 & n == 0:
%o A022341         print(n, end=',')
%o A022341 (Python)
%o A022341 def A022341(n):
%o A022341     tlist, s = [1,2], 0
%o A022341     while tlist[-1]+tlist[-2] <= n: tlist.append(tlist[-1]+tlist[-2])
%o A022341     for d in tlist[::-1]:
%o A022341         if d <= n:
%o A022341             s += 1
%o A022341             n -= d
%o A022341         s <<= 1
%o A022341     return (s<<1)|1 # _Chai Wah Wu_, Apr 24 2025
%o A022341 (Scala) (1 to 511 by 2).filter(n => (n & 2 * n) == 0) // _Alonso del Arte_, Apr 12 2020
%o A022341 (C#)
%o A022341 public static bool IsOddFibbinaryNum(this int n) => ((n & (n >> 1)) == 0) && (n % 2 == 1) ? true : false; // _Frank Hollstein_, Jul 07 2021
%Y A022341 Cf. A000045, A003714, A000846, A022340.
%Y A022341 First column of A356875.
%K A022341 nonn,easy
%O A022341 0,2
%A A022341 _Marc LeBrun_
%E A022341 More terms from _Benoit Cloitre_, May 23 2004 and _Alonso del Arte_, Jun 18 2012
%E A022341 Name edited by _Peter Munn_, Sep 02 2022