A003726 -id:A003726 - cp's OEIS frontend

A319431 The first differences (A319430) of the tribonacci representation numbers (A003726 or A278038) consists of runs of 1's separated by the terms of the present sequence.

2, 3, 2, 5, 2, 3, 10, 2, 3, 2, 5, 2, 19, 2, 3, 2, 5, 2, 3, 10, 2, 3, 2, 37, 2, 3, 2, 5, 2, 3, 10, 2, 3, 2, 5, 2, 19, 2, 3, 2, 5, 2, 3, 74, 2, 3, 2, 5, 2, 3, 10, 2, 3, 2, 5, 2, 19, 2, 3, 2, 5, 2, 3, 10, 2, 3, 2, 37, 2, 3, 2, 5, 2, 3, 10, 2, 3, 2, 5, 2, 147, 2, 3, 2, 5, 2, 3, 10, 2, 3, 2, 5, 2, 19, 2, 3, 2, 5, 2

Offset: 1

N. J. A. Sloane, Sep 30 2018

nonn

The runs of 1's in A319430 have lengths that apparently are given by A275925 (with a slight change at the start). The present sequence shows the terms greater than 1.
Let b = "2,3,2", c = "2,3,2,5,2,3", d = "2,3,2,5,2". The sequence appears to consist of a word over the alphabet {b,c,d} interspersed with the sequence 10, 19, 10, 37, 10, 19, 74, 10, 19, 10, 37, 10, 147, 10, 19, 10, 37, 10, ...:
c, 10, d, 19, c, 10, b, 37, c, 10, d, 19, c, 74, c, 10, d, 19, c, 10, b, 37, c, 10, d, 147, c, 10, d, 19, c, 10, b, 37, c, 10, d, 19, c, 74, ...
The interspersed sequence 10, 19, 10, 37, 10, 19, 74, ... appears to have the same kind of structure.
It would be nice to have a recurrence of some kind that produces this sequence.

			0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 24, 25, ... = A003726 (trib. repres. numbers)
1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, ... = A319430 (differences)
2, 3, 2, 5, 2, 3, 10, 2, 3, 2, 5, 2, 19, 2, 3, 2, 5, 2, 3, 10, 2, 3, 2, 37, 2, ... (omit 1's, present sequence)
6, 1, 5, 1, 6, 1, 3, 1, 6, 1, 5, 1, 6, 1, 6, 1, 5, 1, 6, 1, 3, 1, 6, 1, 5, 1, ... = run lengths in differences
6, 5, 6, 3, 6, 5, 6, 6, 5, 6, 3, 6, 5, 6, 5, 6, 3, 6, 5, 6, 6, 5, 6, 3, 6, 5, 6, ... = A275925 truncated (BISECTION of run lengths)

Rémy Sigrist, Table of n, a(n) for n = 1..50000 (first 1606 terms from N. J. A. Sloane)

Cf. A003726, A275925, A278038, A319430.

DeleteCases[Differences@ Select[Range[0, 1200], SequenceCount[IntegerDigits[#, 2], {1, 1, 1}] == 0 &] , 1] (* Michael De Vlieger, Dec 23 2019 *)

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.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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

			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)

Donald E. Knuth, The Art of Computer Programming: Fundamental Algorithms, Vol. 1, 2nd ed., Addison-Wesley, 1973, pp. 85, 493.

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.
O-Yeat Chan and Dante Manna, Divisibility properties of Stirling numbers of the second kind.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
David Eppstein, Generating fibbinary numbers, three ways, 2021.
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.
Ron Knott, Rabbit Sequence in Zeckendorf Expansion (A003714).
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.
A. J. Macfarlane, On the fibbinary numbers and the Wythoff array, arXiv:2405.18128 [math.CO], 2024. See pages 1, 5.
Douglas M. McKenna, Fibbinary Zippers in a Monoid of Toroidal Hamiltonian Cycles that Generate Hilbert-Style Square-Filling Curves, ECA 2:2 (2022) Article S2R13.
Joris Nieuwveld, Fractions, Functions and Folding. A Novel Link between Continued Fractions, Mahler Functions and Paper Folding, Master's Thesis, arXiv:2108.11382 [math.NT], 2021.
Yaron Shany and Amit Berman, The Generating Idempotent Is a Minimum-Weight Codeword for Some Binary BCH Codes, arXiv:2408.08218 [cs.IT], 2024. See p. 10.
N. J. A. Sloane, Table of n, a(n) (base 10), a(n) (base 2), for n = 0..1000.
Chai Wah Wu, Record values in appending and prepending bitstrings to runs of binary digits, arXiv:1810.02293 [math.NT], 2018.
Index entries for 2-automatic sequences.
Index entries for sequences defined by congruent products between domains N and GF(2)[X].
Index entries for sequences defined by congruent products under XOR.

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.
Cf. A000045, A005203, A005590, A007895, A037011, A048728, A048679, A056017, A060112, A072649, A083368, A089939, A106027, A106028, A116361.
3*a(n) is in A001969.
Cf. also A215022-A215025, A242407.
Cf. A014081 (count 11 bits).

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

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:

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 *)

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

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<A003714(t)) \\ M. F. Hasler, Nov 30 2021

for n in range(300):
    if 2*n & n == 0:
        print(n, end=",") # Alex Ratushnyak, Jun 21 2012

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

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 */

(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

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

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

A003754 Numbers with no adjacent 0's in binary expansion.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 21, 22, 23, 26, 27, 29, 30, 31, 42, 43, 45, 46, 47, 53, 54, 55, 58, 59, 61, 62, 63, 85, 86, 87, 90, 91, 93, 94, 95, 106, 107, 109, 110, 111, 117, 118, 119, 122, 123, 125, 126, 127, 170, 171, 173, 174, 175, 181

Offset: 1

N. J. A. Sloane

Theorem (J.-P. Allouche, J. Shallit, G. Skordev): This sequence = A052499 - 1.
Ahnentafel numbers of ancestors contributing the X-chromosome to a female. A280873 gives the male inheritance. - Floris Strijbos, Jan 09 2017 [Equivalence with this sequence pointed out by John Blythe Dobson, May 09 2018]
The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. This sequence lists all numbers k such that the k-th composition in standard order has no parts greater than two. See the corresponding example below. - Gus Wiseman, Apr 04 2020
The binary representation of a(n+1) has the same string of digits as the lazy Fibonacci (also known as dual Zeckendorf) representation of n that uses 0s and 1s. (The "+1" is essentially an adjustment for the offset of this sequence.) - Peter Munn, Sep 06 2022

			21 is in the sequence because 21 = 10101_2. '10101' has no '00' present in it. - _Indranil Ghosh_, Feb 11 2017
From _Gus Wiseman_, Apr 04 2020: (Start)
The terms together with the corresponding compositions begin:
    0: ()            30: (1,1,1,2)         90: (2,1,2,2)
    1: (1)           31: (1,1,1,1,1)       91: (2,1,2,1,1)
    2: (2)           42: (2,2,2)           93: (2,1,1,2,1)
    3: (1,1)         43: (2,2,1,1)         94: (2,1,1,1,2)
    5: (2,1)         45: (2,1,2,1)         95: (2,1,1,1,1,1)
    6: (1,2)         46: (2,1,1,2)        106: (1,2,2,2)
    7: (1,1,1)       47: (2,1,1,1,1)      107: (1,2,2,1,1)
   10: (2,2)         53: (1,2,2,1)        109: (1,2,1,2,1)
   11: (2,1,1)       54: (1,2,1,2)        110: (1,2,1,1,2)
   13: (1,2,1)       55: (1,2,1,1,1)      111: (1,2,1,1,1,1)
   14: (1,1,2)       58: (1,1,2,2)        117: (1,1,2,2,1)
   15: (1,1,1,1)     59: (1,1,2,1,1)      118: (1,1,2,1,2)
   21: (2,2,1)       61: (1,1,1,2,1)      119: (1,1,2,1,1,1)
   22: (2,1,2)       62: (1,1,1,1,2)      122: (1,1,1,2,2)
   23: (2,1,1,1)     63: (1,1,1,1,1,1)    123: (1,1,1,2,1,1)
   26: (1,2,2)       85: (2,2,2,1)        125: (1,1,1,1,2,1)
   27: (1,2,1,1)     86: (2,2,1,2)        126: (1,1,1,1,1,2)
   29: (1,1,2,1)     87: (2,2,1,1,1)      127: (1,1,1,1,1,1,1)
(End)

Indranil Ghosh, Table of n, a(n) for n = 1..50000 (terms 1..1000 from 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.
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.5.
Andreas M. Hinz and Paul K. Stockmeyer, Precious Metal Sequences and Sierpinski-Type Graphs, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
Tomi Kärki, Anne Lacroix, and Michel Rigo, On the recognizability of self-generating sets, Journal of Integer Sequences, Vol. 13 (2010), Article 10.2.2.
Wikipedia, Ahnentafel.
Witzel, Stefan On panel-regular ~A2 lattices Geom. Dedicata 191, 85-135 (2017).
Index entries for 2-automatic sequences.
Index entries for sequences related to binary expansion of n.

A104326(n) = A007088(a(n)); A023416(a(n)) = A087116(a(n)); A107782(a(n)) = 0; A107345(a(n)) = 1; A107359(n) = a(n+1) - a(n); a(A001911(n)) = A000225(n); a(A000071(n+2)) = A000975(n). - Reinhard Zumkeller, May 25 2005
Cf. A003796 (no 000), A004745 (no 001), A004746 (no 010), A004744 (no 011), A004742 (no 101), A004743 (no 110), A003726 (no 111).
Complement of A004753.
Cf. A023705, A196168, A280873.
Positions of numbers <= 2 in A333766 (see this and A066099 for other sequences about compositions in standard order).
Cf. A318928.

a003754 n = a003754_list !! (n-1)
a003754_list = filter f [0..] where
   f x = x == 0 || x `mod` 4 > 0 && f (x `div` 2)
-- Reinhard Zumkeller, Dec 07 2012, Oct 19 2011

isA003754 := proc(n) local bdgs ; bdgs := convert(n,base,2) ; for i from 2 to nops(bdgs) do if op(i,bdgs)=0 and op(i-1,bdgs)= 0 then return false; end if; end do; return true; end proc:
A003754 := proc(n) option remember; if n= 1 then 0; else for a from procname(n-1)+1 do if isA003754(a) then return a; end if; end do: end if; end proc:
# R. J. Mathar, Oct 23 2010

Select[ Range[0, 200], !MatchQ[ IntegerDigits[#, 2], {_, 0, 0, _}]&] (* Jean-François Alcover, Oct 25 2011 *)
Select[Range[0,200],SequenceCount[IntegerDigits[#,2],{0,0}]==0&] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, May 21 2015 *)

is(n)=n=bitor(n,n>>1)+1; n>>=valuation(n,2); n==1 \\ Charles R Greathouse IV, Feb 06 2017

i=0
while i<=500:
    if "00" not in bin(i)[2:]:
        print(str(i), end=',')
    i+=1 # Indranil Ghosh, Feb 11 2017

Sum_{n>=2} 1/a(n) = 4.356588498070498826084131338899394678478395568880140707240875371925764128502... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 12 2022

Removed "2" from the name, because, for example, one could argue that 10001 has 3 adjacent zeros, not 2. - Gus Wiseman, Apr 04 2020

A278038 Binary vectors not containing three consecutive 1's; or, representation of n in the tribonacci base.

Original entry on oeis.org

0, 1, 10, 11, 100, 101, 110, 1000, 1001, 1010, 1011, 1100, 1101, 10000, 10001, 10010, 10011, 10100, 10101, 10110, 11000, 11001, 11010, 11011, 100000, 100001, 100010, 100011, 100100, 100101, 100110, 101000, 101001, 101010, 101011, 101100, 101101, 110000, 110001, 110010, 110011, 110100, 110101, 110110, 1000000

Offset: 0

N. J. A. Sloane, Nov 16 2016

These are the nonnegative numbers written in the tribonacci numbering system.

			The tribonacci numbers (as in A000073(n), for n >= 3) are 1, 2, 4, 7, 13, 24, 44, 81, ... In terms of this base, 7 is written 1000, 8 is 1001, 11 is 1100, 12 is 1101, 13 is 10000, etc. Zero is 0.

N. J. A. Sloane, Table of n, a(n) for n = 0..25280
L. Carlitz, Richard Scoville, and V. E. Hoggatt, Jr., Fibonacci Representations of Higher Order, Part 2, The Fibonacci Quarterly, Vol. 10, No. 1 (1972), pp. 43-69, 94.
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
Eric Duchêne and Michel Rigo, A morphic approach to combinatorial games: the Tribonacci case. RAIRO - Theoretical Informatics and Applications, 42, 2008, pp 375-393. See Table 2. [Also available from Numdam]
V. E. Hoggatt, Jr. and Marjorie Bicknell-Johnson, Lexicographic Ordering and Fibonacci Representations, The Fibonacci Quarterly, Vol. 20, No. 3 (1982), pp. 193-218.
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.

Cf. A000073, A080843 (tribonacci word, tribonacci tree).
See A003726 for the decimal representations of these binary strings.
Similar sequences: A014417 (Fibonacci), A130310 (Lucas).

# maximum index in A73 such that A73 <= n.
A73floorIdx := proc(n)
    local k ;
    for k from 3 do
        if A000073(k) = n then
            return k ;
        elif A000073(k) > n then
            return k -1 ;
        end if ;
    end do:
end proc:
A278038 := proc(n)
    local k,L,nres ;
    if n = 0 then
        0;
    else
        k := A73floorIdx(n) ;
        L := [1] ;
        nres := n-A000073(k) ;
        while k >= 4 do
            k := k-1 ;
            if nres >= A000073(k) then
                L := [1,op(L)] ;
                nres := nres-A000073(k) ;
            else
                L := [0,op(L)] ;
            end if ;
        end do:
        add( op(i,L)*10^(i-1),i=1..nops(L)) ;
    end if;
end proc:
seq(A278038(n),n=0..40) ; # R. J. Mathar, Jun 08 2022

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; a[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; AppendTo[s, k]; m -= t[k]; k = 1]; FromDigits @ IntegerDigits[Total[2^(s - 1)], 2]]; Array[a, 100, 0] (* Amiram Eldar, Mar 04 2022 *)

A048715 Binary expansion matches (100(0))(0|1|10)?; or, Zeckendorf-like expansion of n using recurrence f(n) = f(n-1) + f(n-3).

Original entry on oeis.org

0, 1, 2, 4, 8, 9, 16, 17, 18, 32, 33, 34, 36, 64, 65, 66, 68, 72, 73, 128, 129, 130, 132, 136, 137, 144, 145, 146, 256, 257, 258, 260, 264, 265, 272, 273, 274, 288, 289, 290, 292, 512, 513, 514, 516, 520, 521, 528, 529, 530, 544, 545, 546, 548, 576, 577, 578, 580

Offset: 0

Antti Karttunen, Mar 30 1999

No more than one 1-bit in each bit triple.
All terms satisfy A048727(n) = 7*n.
Constructed from A000930 in the same way as A003714 is constructed from A000045.
It appears that n is in the sequence if and only if C(7n,n) is odd (cf. A003714). - Benoit Cloitre, Mar 09 2003
The conjecture by Benoit is correct. This is easily proved using the well-known result that the multiplicity with which a prime p divides C(n+m,n) is the number of carries when adding n+m in base p. - Franklin T. Adams-Watters, Oct 06 2009
Appears to be the set of numbers x such that (x AND 5*x) = x and (x OR 3*x)/x = 3. - Gary Detlefs, Jun 08 2024

G. C. Greubel, Table of n, a(n) for n = 0..1275
Sebastian Karlsson, Walnut code that verifies the conjectures of Paul D. Hanna
Walnut can be downloaded from https://cs.uwaterloo.ca/~shallit/walnut.html.
Index entries for sequences defined by congruent products between domains N and GF(2)[X]
Index entries for sequences defined by congruent products under XOR
Index entries for 2-automatic sequences.

Subsequence of A048716.

Cf. A003726, A004742, A004743, A004744, A048717, A048718, A048719, A048730, A048733, A115422, A115423, A115424.

Reap[Do[If[OddQ[Binomial[7n, n]], Sow[n]], {n, 0, 400}]][[2, 1]]
(* Second program: *)
filterQ[n_] := With[{bb = IntegerDigits[n, 2]}, !MatchQ[bb, {_, 1, 0, 1, _}|{_, 1, 1, _}]];
Select[Range[0, 580], filterQ] (* Jean-François Alcover, Dec 31 2020 *)

is(n)=!bitand(n, 6*n) \\ Charles R Greathouse IV, Oct 03 2016

for my $k (0..580) { print "$k, " if sprintf("%b", $k) =~ m{^(100(0)*)*(0|1|10)?$}; } # Georg Fischer, Jun 26 2021

import re
def ok(n): return re.fullmatch('(100(0)*)*(0|1|10)?', bin(n)[2:]) != None
print(list(filter(ok, range(581)))) # Michael S. Branicky, Jun 26 2021

a(0) = 0, a(n) = (2^(invfoo(n)-1))+a(n-foo(invfoo(n))), where foo(n) is foo(n-1) + foo(n-3) (A000930) and invfoo is its "integral" (floored down) inverse.
a(n) XOR 6*a(n) = 7*a(n); 3*a(n) XOR 4*a(n) = 7*a(n); 3*a(n) XOR 5*a(n) = 6*a(n); (conjectures). - Paul D. Hanna, Jan 22 2006
The conjectures can be verified using the Walnut theorem-prover (see links). - Sebastian Karlsson, Dec 31 2022

Definition corrected by Georg Fischer, Jun 26 2021

A003796 Numbers with no 3 adjacent 0's in binary expansion.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92

Offset: 1

N. J. A. Sloane

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.
David Garth and Adam Gouge, Affinely Self-Generating Sets and Morphisms, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.5.
Chai Wah Wu, Record values in appending and prepending bitstrings to runs of binary digits, arXiv:1810.02293 [math.NT], 2018.
Index entries for 2-automatic sequences.

Complement of A004779.
Cf. A004745 (no 001), A004746 (no 010), A004744 (no 011), A003754 (no 100), A004742 (no 101), A004743 (no 110), A003726 (no 111).
Cf. A063037, A007088.

a003796 n = a003796_list !! (n-1)
a003796_list = filter f [0..] where
   f x  = x < 4 || x `mod` 8 /= 0 && f (x `div` 2)
-- Reinhard Zumkeller, Jul 01 2013

Select[Range[0,100],SequenceCount[IntegerDigits[#,2],{0,0,0}]==0&] (* The program uses the SequenceCount function from Mathematica version 10 *) (* Harvey P. Dale, Sep 12 2015 *)

is(n)=while(n>7,if(bitand(n,7)==0,return(0));n>>=1); 1 \\ Charles R Greathouse IV, Feb 11 2017

Sum_{n>=2} 1/a(n) = 9.829256652701616366441622119246549956902006567009112470631751387637507184399... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 13 2022

A048716 Numbers n such that binary expansion matches ((0)00(1?)1)(0*).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 17, 18, 19, 24, 25, 32, 33, 34, 35, 36, 38, 48, 49, 50, 51, 64, 65, 66, 67, 68, 70, 72, 73, 76, 96, 97, 98, 99, 100, 102, 128, 129, 130, 131, 132, 134, 136, 137, 140, 144, 145, 146, 147, 152, 153, 192, 193, 194, 195, 196, 198, 200, 201

Offset: 1

Antti Karttunen, Mar 30 1999

If bit i is 1, then bits i+-2 must be 0. All terms satisfy A048725(n) = 5*n.
It appears that n is in the sequence if and only if C(5n,n) is odd (cf. A003714). - Benoit Cloitre, Mar 09 2003
Yes, as remarked in A048715, "This is easily proved using the well-known result that the multiplicity with which a prime p divides C(n+m,n) is the number of carries when adding n+m in base p." - Jason Kimberley, Dec 21 2011
A116361(a(n)) <= 2. - Reinhard Zumkeller, Feb 04 2006

Superset of A048715 and A048719. Union of A004742 and A003726.

Cf. A048729, A003714, A115845, A115847, A116360.

Reap[Do[If[OddQ[Binomial[5n, n]], Sow[n]], {n, 0, 400}]][[2, 1]]
(* Second program: *)
filterQ[n_] := With[{bb = IntegerDigits[n, 2]}, MatchQ[bb, {0}|{1}|{1, 1}|{_, 0, , 1, __}|{_ 1, , 0, __}] && !MatchQ[bb, {_, 1, , 1, __}]];
Select[Range[0, 201], filterQ] (* Jean-François Alcover, Dec 31 2020 *)

is(n)=!bitand(n,n>>2) \\ Charles R Greathouse IV, Oct 03 2016

list(lim)=my(v=List(),n,t); while(n<=lim, t=bitand(n,n>>2); if(t, n+=1<Charles R Greathouse IV, Oct 22 2021

A063037 Numbers without 3 consecutive equal binary digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 18, 19, 20, 21, 22, 25, 26, 27, 36, 37, 38, 41, 42, 43, 44, 45, 50, 51, 52, 53, 54, 73, 74, 75, 76, 77, 82, 83, 84, 85, 86, 89, 90, 91, 100, 101, 102, 105, 106, 107, 108, 109, 146, 147, 148, 149, 150, 153, 154, 155, 164, 165, 166

Offset: 1

Lior Manor, Jul 05 2001

Complement of A136037; intersection of A003796 and A003726. - Reinhard Zumkeller, Dec 11 2007
From Emeric Deutsch, Jan 27 2018: (Start)
Also 0 together with the indices of the compositions that have no parts larger than 2. For the definition of the index of a composition see A298644.
For example, 105 is in the sequence since its binary form is 1101001 and the composition [2,1,1,2,1] has no parts larger than 2.
On the other hand, 132 is not in the sequence since its binary form is 10000100 and the composition [1,4,1,2] has a part larger than 2.
The command c(n) from the Maple program yields the composition having index n. (End)
The sequence contains A000045(n+1) positive terms with binary length n. - Rémy Sigrist, Sep 30 2022

			The binary representation of 9 (1001) has no 3 consecutive equal digits.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000045, A000975, A166535, A286262.

isA063037 := proc(n)
    local bdgs,rep,d,i ;
    if n = 0 then
        return true;
    end if;
    bdgs := convert(n,base,2) ;
    rep := 1;
    d := op(1,bdgs) ;
    for i from 2 to nops(bdgs) do
        if op(i,bdgs) = op(i-1,bdgs) then
            rep := rep+1 ;
        else
            rep :=1 ;
        end if ;
        if rep > 2 then
            return false;
        end if;
    end do:
    return true ;
end proc:
for n from 0 to 50 do
    if isA063037(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Dec 18 2013
# Second Maple program:
Runs := proc (L) local j, r, i, k: j := 1: r[j] := L[1]: for i from 2 to nops(L) do if L[i] = L[i-1] then r[j] := r[j], L[i] else j := j+1: r[j] := L[i] end if end do: [seq([r[k]], k = 1 .. j)] end proc: RunLengths := proc (L) map(nops, Runs(L)) end proc: c := proc (n) ListTools:-Reverse(convert(n, base, 2)): RunLengths(%) end proc: A := {0}: for n to 250 do if `subset`(convert(c(n), set), {1, 2}) then A := `union`(A, {n}) else end if end do: A; # most of this Maple program is due to W. Edwin Clark. - Emeric Deutsch, Jan 27 2018

Select[Range[0, 168], AllTrue[Length /@ Split@ IntegerDigits[#, 2], # < 3 &] &] (* Michael De Vlieger, Aug 20 2017 *)

{ n=0; for (m=0, 10^9, x=m; t=1; b=2; while (x>0, d=x-2*(x\2); x\=2; if (d==b, c++; if (c==3, t=x=0), b=d; c=1)); if (t, write("b063037.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 16 2009

a(n) = { n--; if (n<=1, return (n), my (s=1); for (i=1, oo, my (f=fibonacci(i+1)); if (nRémy Sigrist, Sep 30 2022

from itertools import count, islice
def A063037_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n: not ('000' in (s:=bin(n)[2:]) or '111' in s), count(max(0,startvalue)))
A063037_list = list(islice(A063037_gen(),20)) # Chai Wah Wu, Oct 04 2022

It appears (but has not yet been proved) that the terms of {a(n)} can be computed recursively as follows. Let {c(i)} be defined for i >= 4 by c(i) = 2c(i-1) + 1, if i is a multiple of 3, else c(i) = 2c(i-1) - 1, with c(4) = 1. I.e., {c(i)} = {1,1,3,5,9,19,37,73,147,...}, for i=4,5,6,... . Let a(1)=1, a(2)=2, a(3)=3. For n > 3, choose k so that F(k)-2 < n <= F(k+1)-2, where F(k) denotes the k-th Fibonacci number (A000045). Then a(n) = c(k) + 2a(F(k)-2) - a(2F(k)-n-3). This has been verified for n up to 1100. - John W. Layman, May 26 2009

Missing "less than" sign supplied in the conjectured recurrence (thanks to Franklin T. Adams-Watters for pointing this out) by John W. Layman, Nov 09 2009

cp's OEIS Frontend

A319430 First differences of the tribonacci representation numbers (A003726 or A278038).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A319431 The first differences (A319430) of the tribonacci representation numbers (A003726 or A278038) consists of runs of 1's separated by the terms of the present sequence.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Python

Python

Python

Scala

Formula

Extensions

A003754 Numbers with no adjacent 0's in binary expansion.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

Extensions

A278038 Binary vectors not containing three consecutive 1's; or, representation of n in the tribonacci base.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A048715 Binary expansion matches (100(0)*)*(0|1|10)?; or, Zeckendorf-like expansion of n using recurrence f(n) = f(n-1) + f(n-3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Perl

A048715 Binary expansion matches (100(0))(0|1|10)?; or, Zeckendorf-like expansion of n using recurrence f(n) = f(n-1) + f(n-3).

A048716 Numbers n such that binary expansion matches ((0)00(1?)1)(0*).