A022342 -id:A022342 - cp's OEIS frontend

A007067 Nearest integer to n*tau where tau = (1+sqrt(5))/2.

0, 2, 3, 5, 6, 8, 10, 11, 13, 15, 16, 18, 19, 21, 23, 24, 26, 28, 29, 31, 32, 34, 36, 37, 39, 40, 42, 44, 45, 47, 49, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 70, 71, 73, 74, 76, 78, 79, 81, 83, 84, 86, 87, 89, 91, 92, 94, 95, 97, 99, 100, 102, 104, 105

Offset: 0

N. J. A. Sloane, Mira Bernstein

First column of inverse Stolarsky array.

A rectangle of size a(n) X n approximates a golden rectangle. So does A295282(n) X n, which targets the golden ratio's underlying objective. These approximations differ first for n = 4 and generally if n = F(6*k)/2, where F(n) = A000045(n) is the n-th Fibonacci number and k >= 1. - Peter Munn, Jan 12 2018

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Iain Fox, Table of n, a(n) for n = 0..10000 (first 1001 terms from Vincenzo Librandi)
Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Classic Sequences.

Cf. A166946 (characteristic function), A007064 (complement).
Different from A026355.
Sequences with similar terms: A022342, A295282.
Other roundings of n*tau: A000201, A004956, A066096.
Cf. A000045 (Fibonacci numbers), A001622 (value of tau).

A007067:=n->round(n*(1+sqrt(5))/2); seq(A007067(n), n=0..100); # Wesley Ivan Hurt, Nov 27 2013

a[n_] := Round[n*GoldenRatio]; Table[a[n], {n, 0, 55}] (* Jean-François Alcover, Jun 27 2012 *)

a(n) = round(n*(1+sqrt(5))/2) \\ Michel Marcus, May 20 2013

from math import isqrt
def A007067(n): return (isqrt(5*n**2<<2)>>1)+n+1>>1 # Chai Wah Wu, Aug 26 2022

Satisfies a(a(n)) = a(n) + n. - Franklin T. Adams-Watters, Aug 14 2006

a(n) = floor((A066096(2*n) + 1)/2). - Peter Munn, Jan 12 2018

A003842 The infinite Fibonacci word: start with 1, repeatedly apply the morphism 1->12, 2->1, take limit; or, start with S(0)=2, S(1)=1, and for n>1 define S(n)=S(n-1)S(n-2), then the sequence is S(oo).

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1

Offset: 0

N. J. A. Sloane

Or, fixed point of the morphism 1->12, 2->1, starting from a(1) = 2.
A Sturmian word, as are all versions of this sequence. This means that if one slides a window of length n along the sequence, one sees exactly n+1 different subwords (see A213975). For a proof, see for example Chap. 2 of Lothaire (2002).
The limiting mean of the first n terms is 3 - phi, where phi is the golden ratio (A001622); the limiting variance is 2 - phi. - Clark Kimberling, Mar 12 2014
The Wikipedia article on L-system Example 1 is "Algae" given by the axiom: A and rules: A -> AB, B -> A. The sequence G(n) = G(n-1)G(n-2) yields this sequence when A -> 1, B -> 2. - Michael Somos, Jan 12 2015
In the limit #1's : #2's = phi : 1. - Frank M Jackson, Mar 12 2018

			Over the alphabet {a,b} this is the sequence a, b, a, a, b, a, b, a, a, b, a, a, b, a, b, a, a, b, a, b, a, a, b, a, a, b, a, b, a, a, b, a, a, b, a, b, a, a, b, a, b, a, a, b, a, a, b, a, b, a, a, b, a, b, a, a, b, a, a, b, a, b, a, a, b, a, a, b, a, b, a, a, b, a, b, a, a, b, a, a, b, a, b, a, a, b, a, a, b, a, b, a, a, b, a, b, a, a, b, a, a, b, a, b, a, ...

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003.
Jean Berstel, "Fibonacci words—a survey." In The book of L, pp. 13-27. Springer Berlin Heidelberg, 1986.
J. Berstel and J. Karhumaki, Combinatorics on words - a tutorial, Bull. EATCS, #79 (2003), pp. 178-228.
E. Bombieri and J. Taylor, Which distribution of matter diffracts? An initial investigation, in International Workshop on Aperiodic Crystals (Les Houches, 1986), J. de Physique, Colloq. C3, 47 (1986), C3-19 to C3-28.
Aldo de Luca and Stefano Varricchio, Finiteness and regularity in semigroups and formal languages. Monographs in Theoretical Computer Science. An EATCS Series. Springer-Verlag, Berlin, 1999. x+240 pp. ISBN: 3-540-63771-0 MR1696498 (2000g:68001). See p. 25.
J. C. Lagarias, Number Theory and Dynamical Systems, pp. 35-72 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc. - see p. 64.
G. Melançon, Factorizing infinite words using Maple, MapleTech journal, vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36.

T. D. Noe, Table of n, a(n) for n=0..10945 (20 iterations)
J.-P. Allouche and M. Mendes France, Automata and Automatic Sequences, in: Axel F. and Gratias D. (eds), Beyond Quasicrystals. Centre de Physique des Houches, vol 3. Springer, Berlin, Heidelberg, pp. 293-367, 1995; DOI https://doi.org/10.1007/978-3-662-03130-8_11.
J.-P. Allouche and M. Mendes France, Automata and Automatic Sequences, in: Axel F. and Gratias D. (eds), Beyond Quasicrystals. Centre de Physique des Houches, vol 3. Springer, Berlin, Heidelberg, pp. 293-367, 1995; DOI https://doi.org/10.1007/978-3-662-03130-8_11. [Local copy]
Scott Balchin and Dan Rust, Computations for Symbolic Substitutions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.1.
Jean Berstel, Home Page
Julien Cassaigne, On extremal properties of the Fibonacci word, RAIRO-Theor. Inf. Appl. 42 (2008) 701-715.
J. Endrullis, D. Hendriks and J. W. Klop, Degrees of streams.
S. Ferenczi, Complexity of sequences and dynamical systems, Discrete Math., 206 (1999), 145-154.
J. Grytczuk, Infinite semi-similar words, Discrete Math. 161 (1996), 133-141.
A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic Schrödinger operators, Commun. Math. Phys. 174 (1995), 149-159.
Clark Kimberling, A Self-Generating Set and the Golden Mean, J. Integer Sequences, 3 (2000), #00.2.8.
Clark Kimberling, Intriguing infinite words composed of zeros and ones, Elemente der Mathematik (2021).
M. Lothaire, Algebraic Combinatorics on Words, Cambridge, 2002, see p. 41, etc.
G. Melançon, Lyndon factorization of sturmian words, Discr. Math., 210 (2000), 137-149.
F. Mignosi, A. Restivo and M. Sciortino, Words and forbidden factors, WORDS (Rouen, 1999). Theoret. Comput. Sci. 273 (2002), no. 1-2, 99--117. MR1872445 (2002m:68096) - From _N. J. A. Sloane_, Jul 10 2012
F. Mignosi and L. Q. Zamboni, On the number of Arnoux-Rauzy words, Acta arith., 101 (2002), no. 2, 121-129.
T. D. Noe, The first 1652 subwords of A003849, including leading zeros.
Giuseppe Pirillo, Fibonacci numbers and words, Discrete Math. 173 (1997), no. 1-3, 197--207. MR1468849 (98g:68135).
Patrice Séébold, Look and Say Fibonacci, RAIRO-Theor. Inf. Appl. 42 (2008) 729-746.
N. J. A. Sloane, The first 10946 terms, concatenated
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Eric Weisstein's World of Mathematics, Golden Ratio
Wikipedia, L-system Example 1: Algae
Index entries for sequences that are fixed points of mappings

A003849 is another common version of this sequence.
See also A014675, A005614, A001468, A106750, A213975, A213976, A214317, A214318, A182028, A120613.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - N. J. A. Sloane, Mar 11 2021

a003842 n = a003842_list !! n
a003842_list = tail $ concat fws where
   fws = [2] : [1] : (zipWith (++) fws $ tail fws)
-- Reinhard Zumkeller, Oct 26 2013

Nest[ Flatten[ # /. {1 -> {1, 2}, 2 -> {1}}] &, {1}, 10] (* Robert G. Wilson v, Mar 04 2005 *)
Table[n + 1 - Floor[((1 + Sqrt[5])/2)*Floor[2*(n + 1)/(1 + Sqrt[5])]], {n, 1, 50}] (* G. C. Greubel, May 18 2017 *)
SubstitutionSystem[{1->{1,2},2->{1}},{1},{10}][[1]] (* Harvey P. Dale, Nov 19 2022 *)

for(n=1,50, print1(n+1 - floor(((1+sqrt(5))/2)*floor(2*(n+1)/(1+sqrt(5)))), ", ")) \\ G. C. Greubel, May 18 2017

def A003842(length):
    a = [1]
    while len(a)Nicholas Stefan Georgescu, Jun 14 2022

def aupto(nn):
    S, Fnm2, Fnm1 = [1, 2], 1, 2
    while len(S) < nn+1:
        S += S[:min(Fnm2, nn+1-len(S))]
        Fnm2, Fnm1 = Fnm1, Fnm1+Fnm2
    return S
print(aupto(104)) # Michael S. Branicky, Jun 06 2022

from math import isqrt
def A003842(n): return n+2-((m:=(n+2+isqrt(5*(n+2)**2)>>1)-n-2)+isqrt(5*m**2)>>1) # Chai Wah Wu, Aug 26 2022

Define strings S(0)=2, S(1)=1, S(n)=S(n-1)S(n-2); iterate. Sequence is S(infinity).

a(n) = n + 2 - A120613(n+1). - Benoit Cloitre, Jul 28 2005 [Corrected by N. J. A. Sloane, Jun 30 2018]

Entry revised by N. J. A. Sloane, Jul 03 2012

A014675 The infinite Fibonacci word (start with 1, apply 1->2, 2->21, take limit).

Original entry on oeis.org

2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2

Offset: 0

N. J. A. Sloane

The limiting mean and variance of the first n terms are both equal to the golden ratio (A001622). - Clark Kimberling, Mar 12 2014
Let F = A000045 (Fibonacci numbers).  For n >= 3, the first F(n)-2 terms of A014675 form a palindrome; see A001911.  If k is not one of the numbers F(n)-2, then the first k terms of A014675 do not form a palindrome. - Clark Kimberling, Jul 14 2014
First differences of A000201. - Tom Edgar, Apr 23 2015 [Editor's note: except for the offset: as for A022342, below. - M. F. Hasler, Oct 13 2017]
Also first differences of A022342 (which starts at offset 1): a(n)=A022342(n+2)-A022342(n+1), n >= 0. Equal to A001468 without its first term: a(n) = A001468(n+1), n >= 0. - M. F. Hasler, Oct 13 2017
The word is a concatenation of three runs: 1, 2, and 22. The limiting proportions of these are respectively 1/2, 1 - phi/2, and (phi - 1)/2, where phi = golden ratio.  The mean runlength is (phi + 1)/2. - Clark Kimberling, Dec 26 2010

D. Gault and M. Clint, "Curiouser and curiouser" said Alice. Further reflections on an interesting recursive function, Internat. J. Computer Math., 26 (1988), 35-43. See Table 2.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7, p. 36.
G. Melançon, Factorizing infinite words using Maple, MapleTech journal, vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36.

T. D. Noe, Table of n, a(n) for n = 0..10945 (20 iterations)
M. Bunder and K. Tognetti, On the self matching properties of [j tau], Discrete Math., 241 (2001), 139-151.
D. Gault and M. Clint, "Curiouser and curiouser said Alice. Further reflections on an interesting recursive function, Intern. J. Computer. Math., 26 (1988), 35-43. (Annotated scanned copy)
J. Grytczuk, Infinite semi-similar words, Discrete Math. 161 (1996), 133-141.
G. Melançon, Lyndon factorization of sturmian words, Discr. Math., 210 (2000), 137-149.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Index entries for sequences that are fixed points of mappings

This is the {2,1} version. The standard form is A003849 (alphabet {0,1}). See also A005614 (alphabet {1,0}), A003842 (alphabet {1,2} instead of {2,1}).
Cf. A082389, A008351, A000045, A001622, A001911.
Equals A001468 except for initial term.
Differs from A025143 in many entries starting at entry 8.
First differences of A000201 and of A022342.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - N. J. A. Sloane, Mar 11 2021

Digits := 50: t := evalf( (1+sqrt(5))/2); A014675 := n->floor((n+2)*t)-floor((n+1)*t);

Nest[ Flatten[ # /. {1 -> 2, 2 -> {2, 1}}] &, {1}, 11] (* Robert G. Wilson v *)
SubstitutionSystem[{1->{2},2->{2,1}},{1},{11}][[1]] (* Harvey P. Dale, Jan 01 2023 *)

first(n)=my(v=[1],u); while(#vCharles R Greathouse IV, Jun 21 2017

apply( {A014675(n,r=quadgen(5)-1)=(n+2)\r-(n+1)\r}, [0..99]) \\ M. F. Hasler, Apr 07 2021, improved on suggestion from Kevin Ryde, Apr 23 2021

from math import isqrt
def A014675(n): return (n+2+isqrt(m:=5*(n+2)**2)>>1)-(n+1+isqrt(m-10*n-15)>>1) # Chai Wah Wu, Aug 10 2022

Define strings S(0)=1, S(1)=2, S(n)=S(n-1).S(n-2) for n>=2. Sequence is S(infinity).

a(n) = floor((n+2)*phi) - floor((n+1)*phi) = A000201(n+2) - A000201(n+1), phi = (1 + sqrt(5))/2.

Corrected by N. J. A. Sloane, Nov 07 2001

A022290 Replace 2^k in binary expansion of n with Fibonacci(k+2).

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 6, 5, 6, 7, 8, 8, 9, 10, 11, 8, 9, 10, 11, 11, 12, 13, 14, 13, 14, 15, 16, 16, 17, 18, 19, 13, 14, 15, 16, 16, 17, 18, 19, 18, 19, 20, 21, 21, 22, 23, 24, 21, 22, 23, 24, 24, 25, 26, 27, 26, 27, 28, 29, 29, 30, 31, 32, 21, 22, 23, 24, 24, 25, 26

Offset: 0

Marc LeBrun

			n=4 = 2^2 is replaced by A000045(2+2) = 3. n=5 = 2^2 + 2^0 is replaced by A000045(2+2) + A000045(0+2) = 3+1 = 4. - _R. J. Mathar_, Jan 31 2015
From _Philippe Deléham_, Jun 05 2015: (Start)
This sequence regarded as a triangle with rows of lengths 1, 1, 2, 4, 8, 16, ...:
  0
  1
  2, 3
  3, 4, 5, 6
  5, 6, 7, 8, 8, 9, 10, 11
  8, 9, 10, 11, 11, 12, 13, 14, 13, 14, 15, 16, 16, 17, 18, 19
  ...
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Other sequences that are built by replacing 2^k in the binary representation with other numbers: A029931 (naturals), A054204 (even-indexed Fibonacci numbers), A062877 (odd-indexed Fibonacci numbers), A059590 (factorials), A089625 (primes).

Cf. A003754, A022342, A026274.

a022290 0 = 0
a022290 n = h n 0 $ drop 2 a000045_list where
   h 0 y _      = y
   h x y (f:fs) = h x' (y + f * r) fs where (x',r) = divMod x 2
-- Reinhard Zumkeller, Oct 03 2012

A022290 := proc(n)
    dgs := convert(n,base,2) ;
    add( op(i,dgs)*A000045(i+1),i=1..nops(dgs)) ;
end proc: # R. J. Mathar, Jan 31 2015
# second Maple program:
b:= (n, i, j)-> `if`(n=0, 0, j*irem(n, 2, 'q')+b(q, j, i+j)):
a:= n-> b(n, 1$2):
seq(a(n), n=0..127);  # Alois P. Heinz, Jan 26 2022

Table[Reverse[#].Fibonacci[1 + Range[Length[#]]] &@ IntegerDigits[n, 2], {n, 0, 54}] (* IWABUCHI Yu(u)ki, Aug 01 2012 *)

my(m=Mod('x,'x^2-'x-1)); a(n) = subst(lift(subst(Pol(binary(n)), 'x,m)), 'x,2); \\ Kevin Ryde, Sep 22 2020

def A022290(n):
    a, b, s = 1,2,0
    for i in bin(n)[-1:1:-1]:
        s += int(i)*a
        a, b = b, a+b
    return s # Chai Wah Wu, Sep 10 2022

G.f.: (1/(1-x)) * Sum_{k>=0} F(k+2)*x^2^k/(1+x^2^k), where F = A000045.
a(n) = Sum_{k>=0} A030308(n,k)*A000045(k+2). - Philippe Deléham, Oct 15 2011
a(A003714(n)) = n. - R. J. Mathar, Jan 31 2015
a(A000225(n)) = A001911(n). - Philippe Deléham, Jun 05 2015
From Jeffrey Shallit, Jul 17 2018: (Start)
Can be computed from the recurrence:
a(4*k)   = a(k) + a(2*k),
a(4*k+1) = a(k) + a(2*k+1),
a(4*k+2) = a(k) - a(2*k) + 2*a(2*k+1),
a(4*k+3) = a(k) - 2*a(2*k) + 3*a(2*k+1),
and the initial terms a(0) = 0, a(1) = 1. (End)
a(A003754(n)) = n-1. - Rémy Sigrist, Jan 28 2020
From Rémy Sigrist, Aug 04 2022: (Start)
Empirically:
- a(2*A003714(n)) = A022342(n+1),
- a(3*A003714(n)) = a(4*A003714(n)) = A026274(n) for n > 0.
(End)

A035336 a(n) = 2floor(nphi) + n - 1, where phi = (1+sqrt(5))/2.

Original entry on oeis.org

2, 7, 10, 15, 20, 23, 28, 31, 36, 41, 44, 49, 54, 57, 62, 65, 70, 75, 78, 83, 86, 91, 96, 99, 104, 109, 112, 117, 120, 125, 130, 133, 138, 143, 146, 151, 154, 159, 164, 167, 172, 175, 180, 185, 188, 193, 198, 201, 206, 209, 214, 219, 222, 227, 230, 235, 240

Offset: 1

N. J. A. Sloane and J. H. Conway

nonn

Second column of Wythoff array.
These are the numbers in A022342 that are not images of another value of the same sequence if it is given offset 0. - Michele Dondi (bik.mido(AT)tiscalenet.it), Dec 30 2001
Also, positions of 2's in A139764, the smallest term in Zeckendorf representation of n. - John W. Layman, Aug 25 2011
From Amiram Eldar, Mar 21 2022: (Start)
Numbers k for which the Zeckendorf representation A014417(k) ends with 0, 1, 0.
The asymptotic density of this sequence is sqrt(5)-2. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J.-P. Allouche and F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018.
Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 4.
J. H. Conway and N. J. A. Sloane, Notes on the Para-Fibonacci and related sequences.
Clark Kimberling, Complementary equations and Wythoff Sequences, JIS, Vol. 11 (2008), Article 08.3.3.
Clark Kimberling, Intriguing infinite words composed of zeros and ones, Elemente der Mathematik (2021).
Clark Kimberling and Kenneth B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, Vol. 123, No. 2 (2016), pp. 267-273.
Johan Kok, Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs, arXiv:2507.16500 [math.CO], 2025. See p. 5.
N. J. A. Sloane, Classic Sequences.

Cf. A001622, A014417, A022342, A066096.
Cf. A139764, A089910, A194584.
Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864.

import Data.List (elemIndices)
a035336 n = a035336_list !! (n-1)
a035336_list = elemIndices 0 a005713_list
-- Reinhard Zumkeller, Dec 30 2011

[2*Floor(n*(1+Sqrt(5))/2)+n-1: n in [1..80]]; // Vincenzo Librandi, Nov 19 2016

Digits := 100: t := (1+sqrt(5))/2; [ seq(2*floor((n+1)*t)+n,n=0..80) ];

Table[2*Floor[n*(1 + Sqrt[5])/2] + n - 1, {n, 50}] (* Wesley Ivan Hurt, Nov 21 2017 *)
Array[2 Floor[# GoldenRatio] + # - 1 &, 60] (* Robert G. Wilson v, Dec 12 2017 *)

from sympy import floor
from mpmath import phi
def a(n): return 2*floor(n*phi) + n - 1 # Indranil Ghosh, Jun 10 2017

from math import isqrt
def A035336(n): return (n+isqrt(5*n**2)&-2)+n-1 # Chai Wah Wu, Aug 17 2022

a(n) = B(A(n)), with A(k)=A000201(k) and B(k)=A001950(k) (Wythoff BA-numbers).
a(n) = A(n) + A(A(n)), with A(A(n))=A003622(n) (Wythoff AA-numbers).
Equals A022342(A003622(n)+1). - Michele Dondi (bik.mido(AT)tiscalenet.it), Dec 30 2001, sequence reference updated by Peter Munn, Nov 23 2017
a(n) = 2*A003622(n) - (n - 1) = A003623(n) - 1. - Franklin T. Adams-Watters, Jun 30 2009
A005713(a(n)) = 0. - Reinhard Zumkeller, Dec 30 2011
a(n) = A089910(n) - 2. - Bob Selcoe, Sep 21 2014

A096270 Fixed point of the morphism 0->01, 1->011.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0

Offset: 0

N. J. A. Sloane, Jun 22 2004

This is another version of the Fibonacci word A005614.
(With offset 1) for k>0, a(ceiling(k*phi^2))=0 and a(floor(k*phi^2))=1 where phi=(1+sqrt(5))/2 is the Golden ratio. - Benoit Cloitre, Apr 01 2006
(With offset 1) for n>1 a(A000045(n)) = (1-(-1)^n)/2.
Equals the Fibonacci word A005614 with an initial zero.
Also the Sturmian word of slope phi (cf. A144595). - N. J. A. Sloane, Jan 13 2009
More precisely: (a(n)) is the inhomogeneous Sturmian word of slope phi-1 and intercept 0: a(n) = floor((n+1)*(phi-1)) - floor(n*(phi-1)), n >= 0. - Michel Dekking, May 21 2018
The ratio of number of 1's to number of 0's tends to the golden ratio (1+sqrt(5))/2 = 1.618... - Zak Seidov, Feb 15 2012

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003.

A.H.M. Smeets, Table of n, a(n) for n = 0..20000
Scott Balchin and Dan Rust, Computations for Symbolic Substitutions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.1.
J-P. Borel and François Laubie, Construction de mots de Christoffel, Comptes rendus de l'Académie des sciences. Série 1, Mathématique 313.8 (1991): 483-485.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Richard Southwell and Jianwei Huang, Complex Networks from Simple Rewrite Systems, arXiv preprint arXiv:1205.0596 [cs.SI], 2012.
Index entries for sequences that are fixed points of mappings

Cf. A003849, A096268, A001519. See A005614, A114986 for other versions.

The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - N. J. A. Sloane, Mar 11 2021

[-1+Floor(n*(1+Sqrt(5))/2)-Floor((n-1)*(1+Sqrt(5))/2): n in [1..100]]; // Wesley Ivan Hurt, Aug 29 2022

Nest[ Function[l, {Flatten[(l /. {0 -> {0, 1}, 1 -> {0, 1, 1}})]}], {0}, 6] (* Robert G. Wilson v, Feb 04 2005 *)

a(n)=-1+floor(n*(1+sqrt(5))/2)-floor((n-1)*(1+sqrt(5))/2) \\ Benoit Cloitre, Apr 01 2006

from math import isqrt
def A096270(n): return (n+1+isqrt(5*(n+1)**2)>>1)-(n+isqrt(5*n**2)>>1)>>1 # Chai Wah Wu, Aug 29 2022

Conjecture: a(n) is given recursively by a(1)=0 and, for n>1, by a(n)=1 if n=F(2k+1) and a(n)=a(n-F(2k+1)) otherwise, where F(2k+1) is the largest odd-indexed Fibonacci number smaller than or equal to n. (This has been confirmed for more than nine million terms.) The odd-indexed bisection of the Fibonacci numbers (A001519) is {1, 2, 5, 13, 34, 89, ...}. So by the conjecture, we would expect that a(30) = a(30-13) = a(17) = a(17-13) = a(4) = a(4-2) = a(2) = 1, which is in fact correct. - John W. Layman, Jun 29 2004
From Michel Dekking, Apr 13 2016: (Start)
Proof of the above conjecture:
Let g be the morphism above: g(0)=01, g(1)=011. Then g^n(0) has length F(2n+1), and (a(n)) starts with g^n(0) for all n>0. Obviously g^n(0) ends in 1 for all n, proving the first part of the conjecture.
We extend the semigroup of words with letters 0 and 1 to the free group, adding the inverses 0*:=0^{-1} and 1*:=1^{-1}. Easy observation: for any word w one has g(w1)= g(w0)1. We claim that for all n>1 one has g^n(0)=u(n)v(n)v(n)0*1, where u(n)=g(u(n-1))0 and v(n)=0*g(v(n-1))0. The recursion starts with u(2)=0, v(2)=10. Indeed: g^2(0)=01011=u(2)v(2)v(2)0*1. Induction step:
  g^{n+1}(0)=g(g^n(0))= g(u(n)v(n)v(n)0*1)= g(u(n)v(n)v(n))1= g(u(n))00*g(v(n))00*g(v(n))00*1=u(n+1)v(n+1)v(n+1)0*1.
Since v(n) has length F(2n-1), which is the largest odd-indexed Fibonacci number smaller than or equal to m for all m between F(2n-1) and F(2n+1), the claim proves the second part of the conjecture. (End)
(With offset 1) a(n) = -1 + floor(n*phi) - floor((n-1)*phi) where phi=(1+sqrt(5))/2 so a(n) = -1 + A082389(n). - Benoit Cloitre, Apr 01 2006

More terms from John W. Layman, Jun 29 2004

A001468 There are a(n) 2's between successive 1's.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2

Offset: 0

N. J. A. Sloane

The Fibonacci word on the alphabet {2,1}, with an extra 1 in front. - Michel Dekking, Nov 26 2018
Start with 1, apply 1->12, 2->122, take limit. - Philippe Deléham, Sep 23 2005
Also number of occurrences of n in Hofstadter G-sequence (A005206) and in A019446. - Reinhard Zumkeller, Feb 02 2012, Aug 07 2011
A block-fractal sequence: every block occurs infinitely many times.  Also a reverse block-fractal sequence.  See A280511. - Clark Kimberling, Jan 06 2017

D. Gault and M. Clint, "Curiouser and curiouser" said Alice. Further reflections on an interesting recursive function, Internat. J. Computer Math., 26 (1988), 35-43. See Table 2.
D. R. Hofstadter, personal communication, Jul 15 1977.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
M. Bunder and K. Tognetti, On the self matching properties of [j tau], Discrete Math., 241 (2001), 139-151.
D. Gault and M. Clint, "Curiouser and curiouser" said Alice. Further reflections on an interesting recursive function, Internat. J. Computer Math., 26 (1988), 35-43.
D. Gault and M. Clint, "Curiouser and curiouser said Alice. Further reflections on an interesting recursive function, Intern. J. Computer. Math., 26 (1988), 35-43. (Annotated scanned copy)
D. R. Hofstadter, Eta-Lore [Cached copy, with permission]
D. R. Hofstadter, Pi-Mu Sequences [Cached copy, with permission]
D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991 [See DRH latter, p. 2, Eq. (2), the sequence marked A006336, which is now A001468].
J. V. Pennington and T. F. Mulcrone, Problem E1226, Amer. Math. Monthly, 64 (1957), 197-198.
Leon Recht, Martin Rosenbaum and E. P. Starke, Problem 4247, Amer. Math. Monthly, 55 (1948), 588-592.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Same as A014675 if initial 1 is deleted. Cf. A003849, A000201, A280511.

import Data.List (group)
a001468 n = a001468_list !! n
a001468_list = map length $ group a005206_list
-- Reinhard Zumkeller, Aug 07 2011

Digits := 100: t := evalf( (1+sqrt(5))/2); A001468 := n-> floor((n+1)*t)-floor(n*t);

Table[Floor[GoldenRatio*(n + 1)] - Floor[GoldenRatio*n], {n, 0, 80}] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006 *)
Nest[ Flatten[# /. {1 -> {1, 2}, 2 -> {1, 2, 2}}] &, {1}, 6] (* Robert G. Wilson v, May 20 2014 and corrected Apr 24 2017 following Clark Kimberling's email of Mar 22 2017 *)
SubstitutionSystem[{1->{1,2},2->{1,2,2}},{1},{6}][[1]] (* Harvey P. Dale, Jan 31 2022 *)

a=[1];for(i=1,30,a=concat([a,vector(a[i],j,2),1]));a \\ Or compute as A001468(n)=A201(n+1)-A201(n) with A201(n)=(n+sqrtint(5*n^2))\2, working for n>=0 although A000201 is defined for n>=1. - M. F. Hasler, Oct 13 2017

def A001468(length):
    a = [1]
    for i in range(length):
        for _ in range(a[i]):
            a.append(2)
        a.append(1)
        if len(a)>=length:
            break
    return a[:length] # Nicholas Stefan Georgescu, Jun 02 2022

from math import isqrt
def A001468(n): return (n+1+isqrt(m:=5*(n+1)**2)>>1)-(n+isqrt(m-10*n-5)>>1) # Chai Wah Wu, Aug 25 2022

a(n) = [(n+1) tau] - [n tau], tau = (1 + sqrt 5)/2 = A001622, [] = floor function.

a(n) = A000201(n+1) - A000201(n) = A022342(n+1) - A022342(n), n >= 1; i.e., the first term discarded, this yields the first differences of A000201 and A022342. - M. F. Hasler, Oct 13 2017

Rechecked by N. J. A. Sloane, Nov 07 2001

A066096 a(n) = floor(n*phi), where phi = (1 + sqrt(5))/2.

Original entry on oeis.org

0, 1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 27, 29, 30, 32, 33, 35, 37, 38, 40, 42, 43, 45, 46, 48, 50, 51, 53, 55, 56, 58, 59, 61, 63, 64, 66, 67, 69, 71, 72, 74, 76, 77, 79, 80, 82, 84, 85, 87, 88, 90, 92, 93, 95, 97, 98, 100, 101, 103, 105, 106

Offset: 0

Michele Dondi (bik.mido(AT)tiscalenet.it), Dec 30 2001

a(n) is the smallest number different from a(i) and a(i)+i for i < n.

The losing positions in the game of Wythoff-Nim are precisely the pairs (a(n), a(n)+n).

G. C. Greubel, Table of n, a(n) for n = 0..10000

Essentially the partial sums of A001468.
Cf. A000201, A001622, A003622, A022342, A026351, A035336, A060143.
Cf. A004919, A004920, A004921, A004922, A004923, A004924, A004925.
Cf. A004926, A004927, A004928, A004929, A004930, A004931, A004932.
Cf. A004933, A004934, A004935, A004976, A090909.

[Floor((1+Sqrt(5))*n/2): n in [0..80]]; // G. C. Greubel, Sep 12 2023

Floor[GoldenRatio*Range[0, 80]] (* G. C. Greubel, Sep 12 2023 *)

a(n) = (n+sqrtint(5*n^2))\2;
[a(n)|n<-[0..100]] \\ Simon Strandgaard, Jun 28 2022

[floor(golden_ratio*n) for n in range(81)] # G. C. Greubel, Sep 12 2023

For n >= 1, a(n) = A000201(n).
Duplicate values in A060143.
a(n) = 1 + A022342(n) = A000201(n).
a(n) = floor(n*phi), where phi = (1 + sqrt(5))/2. - Peter Munn, Jan 12 2018
a(n) = A026351(n) - 1. - Philippe Deléham, Jan 15 2023

Name corrected by Peter Munn, Dec 06 2017

New name using a formula from Peter Munn by Peter Luschny, Jan 18 2023

A001030 Fixed under 1 -> 21, 2 -> 211.

Original entry on oeis.org

2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2

Offset: 1

N. J. A. Sloane

If treated as the terms of a continued fraction, it converges to approximately
2.57737020881617828717350576260723346479894963737498275232531856357441\
7024804797827856956758619431996. - Peter Bertok (peter(AT)bertok.com), Nov 27 2001
There are a(n) 1's between successive 2's. - Eric Angelini, Aug 19 2008
Same sequence where 1's and 2's are exchanged: A001468. - Eric Angelini, Aug 19 2008

Midhat J. Gazale, Number: From Ahmes to Cantor, Section on 'Cleavages' in Chapter 6, Princeton University Press, Princeton, NJ 2000, pp. 203-211.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..8119
N. G. de Bruijn, Sequences of zeros and ones generated by special production rules, Indag. Math., 43 (1981), 27-37.
D. R. Hofstadter, Eta-Lore [Cached copy, with permission]
D. R. Hofstadter, Pi-Mu Sequences [Cached copy, with permission]
D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991
A. Nagel, A self-defining infinite sequence, with an application to Markoff chains and probability, Math. Mag., 36 (1963), 179-183.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282).
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence).

Length of the sequence after 'n' substitution steps is given by the terms of A000129.
Equals A004641(n) + 1.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - N. J. A. Sloane, Mar 11 2021

Following Spage's PARI program.
a001030 n = a001030_list !! (n-1)
a001030_list = [2, 1, 1, 2] ++ f [2] [2, 1, 1, 2] where
   f us vs = ws ++ f vs (vs ++ ws) where
             ws = 1 : us ++ 1 : vs
-- Reinhard Zumkeller, Aug 04 2014

('n' is the number of substitution steps to perform.) Nest[Flatten[ # /. {1 -> {2, 1}, 2 -> {2, 1, 1}}] &, {1}, n]
SubstitutionSystem[{1->{2,1},2->{2,1,1}},{2},{6}][[1]] (* Harvey P. Dale, Feb 15 2022 *)

/* Fast string concatenation method giving e.g. 5740 terms in 8 iterations */
a="2";b="2,1,1,2";print1(b);for(x=1,8,c=concat([",1,",a,",1,",b]);print1(c);a=b;b=concat(b,c)) \\ K. Spage, Oct 08 2009

from math import isqrt
def A001030(n): return [2, 1, 1, 2, 1, 2, 1, 2][n-1] if n < 9 else -isqrt(m:=(n-9)*(n-9)<<1)+isqrt(m+(n-9<<2)+2) # Chai Wah Wu, Aug 25 2022

a(n) = -1 + floor(n*(1+sqrt(2))+1/sqrt(2))-floor((n-1)*(1+sqrt(2))+1/sqrt(2)). - Benoit Cloitre, Jun 26 2004. [I don't know if this is a theorem or a conjecture. - N. J. A. Sloane, May 14 2008]

This is a theorem, following from Hofstadter's Generalized Fundamental Theorem of eta-sequences on page 10 of Eta-Lore. See also de Bruijn's paper from 1981 (hint from Benoit Cloitre). - Michel Dekking, Jan 22 2017

More terms from Peter Bertok (peter(AT)bertok.com), Nov 27 2001

A004641 Fixed under 0 -> 10, 1 -> 100.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1

Offset: 1

N. J. A. Sloane

Partial sums: A088462. - Reinhard Zumkeller, Dec 05 2009

Write w(n) = a(n) for n >= 1. Each w(n) is generated by w(i) for exactly one i <= n; let g(n) = i. Each w(i) generates a single 1, in a word (10 or 100) that starts with 1. Therefore, g(n) is the number of 1s among w(1), ..., w(n), so that g = A088462. That is, this sequence is generated by its partial sums. - Clark Kimberling, May 25 2011

T. D. Noe, Table of n, a(n) for n = 1..8119
Wieb Bosma, Michel Dekking, and Wolfgang Steiner, A remarkable sequence related to Pi and sqrt(2), arXiv:1710.01498 [math.NT], 2017.
Wieb Bosma, Michel Dekking, and Wolfgang Steiner, A remarkable sequence related to Pi and sqrt(2), Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A4.
N. G. de Bruijn, Sequences of zeros and ones generated by special production rules, Nederl. Akad. Wetensch. Indag. Math. 43 (1981), no. 1, 27-37. Reprinted in Physics of Quasicrystals, ed. P. J. Steinhardt et al., p. 664.
C. J. Glasby, S. P. Glasby, and F. Pleijel, Worms by number, Proc. Roy. Soc. B, Proc. Biol. Sci. 275 (1647) (2008) 2071-2076.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence).
Index entries for characteristic functions

Equals A001030 - 1. Essentially the same as A006337 - 1 and A159684.
Characteristic function of A086377.
Cf. A081477.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - N. J. A. Sloane, Mar 11 2021

[Floor(n*(Sqrt(2) - 1) + Sqrt(1/2)) - Floor((n - 1)*(Sqrt(2) - 1) + Sqrt(1/2)): n in [0..100]]; // Vincenzo Librandi, Mar 27 2015

P(0):= (1,0): P(1):= (1,0,0):
((P~)@@6)([1]);
# in Maple 12 or earlier, comment the above line and uncomment the following:
# (curry(map,P)@@6)([1]); # Robert Israel, Mar 26 2015

Nest[ Flatten[# /. {0 -> {1, 0}, 1 -> {1, 0, 0}}] &, {1}, 5] (* Robert G. Wilson v, May 25 2011 *)
SubstitutionSystem[{0->{1,0},1->{1,0,0}},{1},5]//Flatten (* Harvey P. Dale, Nov 20 2021 *)

from math import isqrt
def A004641(n): return [1, 0, 0, 1, 0, 1, 0, 1][n-1] if n < 9 else -1-isqrt(m:=(n-9)*(n-9)<<1)+isqrt(m+(n-9<<2)+2) # Chai Wah Wu, Aug 25 2022

a(n) = floor(n*(sqrt(2) - 1) + sqrt(1/2)) - floor((n - 1)*(sqrt(2) - 1) + sqrt(1/2)) (from the de Bruijn reference). - Peter J. Taylor, Mar 26 2015
From Jianing Song, Jan 02 2019: (Start)
a(n) = A001030(n) - 1.
a(n) = A006337(n-9) - 1 = A159684(n-10) for n >= 10. (End)

cp's OEIS Frontend

A007067 Nearest integer to n*tau where tau = (1+sqrt(5))/2.

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A003842 The infinite Fibonacci word: start with 1, repeatedly apply the morphism 1->12, 2->1, take limit; or, start with S(0)=2, S(1)=1, and for n>1 define S(n)=S(n-1)S(n-2), then the sequence is S(oo).

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A014675 The infinite Fibonacci word (start with 1, apply 1->2, 2->21, take limit).

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A022290 Replace 2^k in binary expansion of n with Fibonacci(k+2).

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A035336 a(n) = 2*floor(n*phi) + n - 1, where phi = (1+sqrt(5))/2.

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A096270 Fixed point of the morphism 0->01, 1->011.

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A035336 a(n) = 2floor(nphi) + n - 1, where phi = (1+sqrt(5))/2.