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A003714 Fibbinary numbers: if n = F(i1) + F(i2) + ... + F(ik) is the Zeckendorf representation of n (i.e., write n in Fibonacci number system) then a(n) = 2^(i1 - 2) + 2^(i2 - 2) + ... + 2^(ik - 2). Also numbers whose binary representation contains no two adjacent 1's. 207
0, 1, 2, 4, 5, 8, 9, 10, 16, 17, 18, 20, 21, 32, 33, 34, 36, 37, 40, 41, 42, 64, 65, 66, 68, 69, 72, 73, 74, 80, 81, 82, 84, 85, 128, 129, 130, 132, 133, 136, 137, 138, 144, 145, 146, 148, 149, 160, 161, 162, 164, 165, 168, 169, 170, 256, 257, 258, 260, 261, 264 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
The name "Fibbinary" is due to Marc LeBrun.
"... integers whose binary representation contains no consecutive ones and noticed that the number of such numbers with n bits was fibonacci(n)". [posting to sci.math by Bob Jenkins (bob_jenkins(AT)burtleburtle.net), Jul 17 2002]
From Benoit Cloitre, Mar 08 2003: (Start)
A number m is in the sequence if and only if C(3m, m) (or equally, C(3m, 2m)) is odd.
a(n) == A003849(n) (mod 2). (End)
Numbers m such that m XOR 2*m = 3*m. - Reinhard Zumkeller, May 03 2005. [This implies that A003188(2*a(n)) = 3*a(n) holds for all n.]
Numbers whose base-2 representation contains no two adjacent ones. For example, m = 17 = 10001_2 belongs to the sequence, but m = 19 = 10011_2 does not. - Ctibor O. Zizka, May 13 2008
m is in the sequence if and only if the central Stirling number of the second kind S(2*m, m) = A007820(m) is odd. - O-Yeat Chan (math(AT)oyeat.com), Sep 03 2009
A000120(3*a(n)) = 2*A000120(a(n)); A002450 is a subsequence.
Every nonnegative integer can be expressed as the sum of two terms of this sequence. - Franklin T. Adams-Watters, Jun 11 2011
Subsequence of A213526. - Arkadiusz Wesolowski, Jun 20 2012
This is also the union of A215024 and A215025 - see the Comment in A014417. - N. J. A. Sloane, Aug 10 2012
The binary representation of each term m contains no two adjacent 1's, so we have (m XOR 2m XOR 3m) = 0, and thus a two-player Nim game with three heaps of (m, 2m, 3m) stones is a losing configuration for the first player. - V. Raman, Sep 17 2012
Positions of zeros in A014081. - John Keith, Mar 07 2022
These numbers are similar to Fibternary numbers A003726, Tribbinary numbers A060140 and Tribternary numbers. This sequence is a subsequence of Fibternary numbers A003726. The number of Fibbinary numbers less than any power of two is a Fibonacci number. We can generate this sequence recursively: start with 0 and 1; then, if x is in the sequence add 2x and 4x+1 to the sequence. The Fibbinary numbers have the property that the n-th Fibbinary number is even if the n-th term of the Fibonacci word is a. Respectively, the n-th Fibbinary number is odd (of the form 4x+1) if the n-th term of the Fibonacci word is b. Every number has a Fibbinary multiple. - Tanya Khovanova and PRIMES STEP Senior, Aug 30 2022
This is the ordered set S of numbers defined recursively by: 0 is in S; if x is in S, then 2*x and 4*x + 1 are in S. See Kimberling (2006) Example 3, in references below. - Harry Richman, Jan 31 2024
REFERENCES
Donald E. Knuth, The Art of Computer Programming: Fundamental Algorithms, Vol. 1, 2nd ed., Addison-Wesley, 1973, pp. 85, 493.
LINKS
G. C. Greubel and T. D. Noe, Table of n, a(n) for n = 0..5000 (terms 0 to 1363 by T. D. Noe)
J.-P. Allouche, J. Shallit and G. Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math., Vol. 292, No. 1-3 (2005), pp. 1-15.
Joerg Arndt, Matters Computational (The Fxtbook), pp. 74-77, pp. 754-756.
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Marc Chamberland and Karl Dilcher, A Binomial Sum Related to Wolstenholme's Theorem, J. Number Theory, Vol. 171, Issue 11 (Nov. 2009), pp. 2659-2672. See Lemma 4.2 p. 2668.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
Clark Kimberling, Affinely recursive sets and orderings of languages, Discrete Math., Vol. 274, No. 1-3 (2004), pp. 147-160.
Roman S. Klyujkov, Quick Fibbinary Numbers Addition, C++ function with test program.
Linus Lindroos, Andrew Sills and Hua Wang, Odd fibbinary numbers and the golden ratio, Fib. Q., Vol. 52, No. 1 (2014), pp. 61-65; alternative link.
FORMULA
No two adjacent 1's in binary expansion.
Let f(x) := Sum_{n >= 0} x^Fibbinary(n). (This is the generating function of the characteristic function of this sequence.) Then f satisfies the functional equation f(x) = x*f(x^4) + f(x^2).
a(0) = 0, a(1) = 1, a(2) = 2, a(n) = 2^(A072649(n) - 1) + a(n - A000045(1 + A072649(n))). - Antti Karttunen
It appears that this sequence gives m such that A082759(3*m) is odd; or, probably equivalently, m such that A037011(3*m) = 1. - Benoit Cloitre, Jun 20 2003
If m is in the sequence then so are 2*m and 4*m + 1. - Henry Bottomley, Jan 11 2005
A116361(a(n)) <= 1. - Reinhard Zumkeller, Feb 04 2006
A085357(a(n)) = 1; A179821(a(n)) = a(n). - Reinhard Zumkeller, Jul 31 2010
a(n)/n^k is bounded (but does not tend to a limit), where k = 1.44... = A104287. - Charles R Greathouse IV, Sep 19 2012
a(n) = a(A193564(n+1))*2^(A003849(n) + 1) + A003849(n) for n > 0. - Daniel Starodubtsev, Aug 05 2021
There are Fibonacci(n+1) terms with up to n bits in this sequence. - Charles R Greathouse IV, Oct 22 2021
Sum_{n>=1} 1/a(n) = 3.704711752910469457886531055976801955909489488376627037756627135425780134020... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 12 2022
EXAMPLE
From Joerg Arndt, Jun 11 2011: (Start)
In the following, dots are used for zeros in the binary representation:
a(n) binary(a(n)) n
0: ....... 0
1: ......1 1
2: .....1. 2
4: ....1.. 3
5: ....1.1 4
8: ...1... 5
9: ...1..1 6
10: ...1.1. 7
16: ..1.... 8
17: ..1...1 9
18: ..1..1. 10
20: ..1.1.. 11
21: ..1.1.1 12
32: .1..... 13
33: .1....1 14
34: .1...1. 15
36: .1..1.. 16
37: .1..1.1 17
40: .1.1... 18
41: .1.1..1 19
42: .1.1.1. 20
64: 1...... 21
65: 1.....1 22
(End)
MAPLE
A003714 := proc(n)
option remember;
if n < 3 then
n ;
else
2^(A072649(n)-1) + procname(n-combinat[fibonacci](1+A072649(n))) ;
end if;
end proc:
seq(A003714(n), n=0..10) ;
# To produce a table giving n, a(n) (base 10), a(n) (base 2) - from N. J. A. Sloane, Sep 30 2018
# binary: binary representation of n, in human order
binary:=proc(n) local t1, L;
if n<0 then ERROR("n must be nonnegative"); fi;
if n=0 then return([0]); fi;
t1:=convert(n, base, 2); L:=nops(t1);
[seq(t1[L+1-i], i=1..L)];
end;
for n from 0 to 100 do t1:=A003714(n); lprint(n, t1, binary(t1)); od:
MATHEMATICA
fibBin[n_Integer] := Block[{k = Ceiling[Log[GoldenRatio, n Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; FromDigits[fr, 2]]; Table[fibBin[n], {n, 0, 61}] (* Robert G. Wilson v, Sep 18 2004 *)
Select[Range[0, 270], ! MemberQ[Partition[IntegerDigits[#, 2], 2, 1], {1, 1}] &] (* Harvey P. Dale, Jul 17 2011 *)
Select[Range[256], BitAnd[#, 2 #] == 0 &] (* Alonso del Arte, Jun 18 2012 *)
With[{r = Range[10^5]}, Pick[r, BitAnd[r, 2 r], 0]] (* Eric W. Weisstein, Aug 18 2017 *)
Select[Range[0, 299], SequenceCount[IntegerDigits[#, 2], {1, 1}] == 0 &] (* Requires Mathematica version 10 or later. -- Harvey P. Dale, Dec 06 2018 *)
PROG
(Haskell)
import Data.Set (Set, singleton, insert, deleteFindMin)
a003714 n = a003714_list !! n
a003714_list = 0 : f (singleton 1) where
f :: Set Integer -> [Integer]
f s = m : (f $ insert (4*m + 1) $ insert (2*m) s')
where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 03 2012, Feb 07 2012
(PARI) msb(n)=my(k=1); while(k<=n, k<<=1); k>>1
for(n=1, 1e4, k=bitand(n, n<<1); if(k, n=bitor(n, msb(k)-1), print1(n", "))) \\ Charles R Greathouse IV, Jun 15 2011
(PARI) select( is_A003714(n)=!bitand(n, n>>1), [0..266])
{(next_A003714(n, t)=while(t=bitand(n+=1, n<<1), n=bitor(n, 1<<exponent(t)-1)); n); } t=0; vector(60, i, t=next_A003714(t)) \\ M. F. Hasler, Nov 30 2021
(Python)
for n in range(300):
if 2*n & n == 0:
print(n, end=", ") # Alex Ratushnyak, Jun 21 2012
(Python)
def A003714(n):
tlist, s = [1, 2], 0
while tlist[-1]+tlist[-2] <= n:
tlist.append(tlist[-1]+tlist[-2])
for d in tlist[::-1]:
s *= 2
if d <= n:
s += 1
n -= d
return s # Chai Wah Wu, Jun 14 2018
(Python)
def fibbinary():
x = 0
while True:
yield x
y = ~(x >> 1)
x = (x - y) & y # Falk Hüffner, Oct 23 2021
(C++)
/* start with x=0, then repeatedly call x=next_fibrep(x): */
ulong next_fibrep(ulong x)
{
// 2 examples: // ex. 1 // ex.2
// // x == [*]0 010101 // x == [*]0 01010
ulong y = x | (x>>1); // y == [*]? 011111 // y == [*]? 01111
ulong z = y + 1; // z == [*]? 100000 // z == [*]? 10000
z = z & -z; // z == [0]0 100000 // z == [0]0 10000
x ^= z; // x == [*]0 110101 // x == [*]0 11010
x &= ~(z-1); // x == [*]0 100000 // x == [*]0 10000
return x;
}
/* Joerg Arndt, Jun 22 2012 */
(Scala) (0 to 255).filter(n => (n & 2 * n) == 0) // Alonso del Arte, Apr 12 2020
(C#)
public static bool IsFibbinaryNum(this int n) => ((n & (n >> 1)) == 0) ? true : false; // Frank Hollstein, Jul 07 2021
CROSSREFS
A007088(a(n)) = A014417(n) (same sequence in binary). Complement: A004780. Char. function: A085357. Even terms: A022340, odd terms: A022341. First difference: A129761.
Other sequences based on similar restrictions on binary expansion: A003726 & A278038, A003754, A048715, A048718, A107907, A107909.
3*a(n) is in A001969.
Cf. A014081 (count 11 bits).
Sequence in context: A339906 A359267 A279430 * A340956 A010402 A010443
KEYWORD
nonn,nice,easy,look
AUTHOR
EXTENSIONS
Edited by Antti Karttunen, Feb 21 2006
Cross reference to A007820 added (into O-Y.C. comment) by Jason Kimberley, Sep 14 2009
Typo corrected by Jeffrey Shallit, Sep 26 2014
STATUS
approved

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Last modified February 27 21:03 EST 2024. Contains 370378 sequences. (Running on oeis4.)