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A022341
a(n) = 4*A003714(n) + 1; the odd Fibbinary numbers.
6
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
0,2
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
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).
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 *)
PROG
(Python)
for n in range(1, 700, 2):
if n*2 & n == 0:
print(n, end=', ')
(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
CROSSREFS
First column of A356875.
Sequence in context: A097538 A001771 A288448 * A255651 A216877 A350989
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Benoit Cloitre, May 23 2004 and Alonso del Arte, Jun 18 2012
Name edited by Peter Munn, Sep 02 2022
STATUS
approved