cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A214207 Replace the terms of A213975 with their ranks in A007931.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 11, 12, 17, 19, 20, 24, 25, 35, 36, 40, 41, 49, 51, 72, 73, 81, 83, 99, 100, 104, 145, 147, 163, 164, 168, 200, 201, 209, 292, 296, 328, 329, 337, 401, 403, 419, 420, 585, 593, 657, 659, 675, 676, 804, 808, 840, 841, 1171, 1187, 1188, 1316, 1320, 1352, 1353, 1609, 1617, 1681, 1683, 2344, 2376, 2377, 2633, 2641, 2705, 2707, 3219, 3235
Offset: 1

Views

Author

N. J. A. Sloane, Jul 09 2012

Keywords

Comments

Created with the hope of obtaining a better understanding of A213975. It would be interesting if this sequence or A214208 had an alternative definition.

Examples

			A213975(10)=1121, which is A007931(17), so a(10)=17.

Links

Crossrefs

Cf. A213975, A007931, A214208, A214209.

A003849 The infinite Fibonacci word (start with 0, apply 0->01, 1->0, take limit).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

A Sturmian word.
Define strings S(0)=0, S(1)=01, S(n)=S(n-1)S(n-2); iterate; sequence is S(infinity). If the initial 0 is omitted from S(n) for n>0, we obtain A288582(n+1).
The 0's occur at positions in A022342 (i.e., A000201 - 1), the 1's at positions in A003622.
Replace each run (1;1) with (1;0) in the infinite Fibonacci word A005614 (and add 0 as prefix) A005614 begins: 1,0,1,1,0,1,0,1,1,0,1,1,... changing runs (1,1) with (1,0) produces 1,0,0,1,0,1,0,0,1,0,0,1,... - Benoit Cloitre, Nov 10 2003
Characteristic function of A003622. - Philippe Deléham, May 03 2004
The fraction of 0's in the first n terms approaches 1/phi (see for example Allouche and Shallit). - N. J. A. Sloane, Sep 24 2007
The limiting mean and variance of the first n terms are 2-phi and 2*phi-3, respectively. - Clark Kimberling, Mar 12 2014, Aug 16 2018
Let S(n) be defined as above. Then this sequence is S(1) + Sum_{n=0..} S(n), where the addition of strings represents concatenation. - Isaac Saffold, May 03 2019
The word is a concatenation of three runs: 0, 1, and 00. The limiting proportions of these are respectively 1 - phi/2, 1/2, and (phi - 1)/2. The mean runlength is (phi + 1)/2. - Clark Kimberling, Dec 26 2010
From Amiram Eldar, Mar 10 2021: (Start)
a(n) is the number of the trailing 0's in the dual Zeckendorf representation of (n+1) (A104326).
The asymptotic density of the occurrences of k (0 or 1) is 1/phi^(k+1), where phi is the golden ratio (A001622).
The asymptotic mean of this sequence is 1/phi^2 (A132338). (End)

Examples

			The word is 010010100100101001010010010100...
Over the alphabet {a,b} this is 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, ...

References

  • 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. 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.
  • Wolfdieter Lang, The Wythoff and the Zeckendorf representations of numbers are equivalent, in G. E. Bergum et al. (edts.) Application of Fibonacci numbers vol. 6, Kluwer, Dordrecht, 1996, pp. 319-337. [See A317208 for a link.]
  • G. Melançon, Factorizing infinite words using Maple, MapleTech journal, vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36.
  • Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

Links

Crossrefs

There are several versions of this sequence in the OEIS. This one and A003842 are probably the most important. See also A008352, A076662, A288581, A288582.
Binary complement of A005614. Cf. A014675, A036299, A003714, A014417, A096268, A096270, A133235, A182028, A213975.
Positions of 1's gives A003622.
Sequences mentioned in the Allouche et al. "Taxonomy" paper, listed by example number: 1: A003849, 2: A010060, 3: A010056, 4: A020985 and A020987, 5: A191818, 6: A316340 and A273129, 18: A316341, 19: A030302, 20: A063438, 21: A316342, 22: A316343, 23: A003849 minus its first term, 24: A316344, 25: A316345 and A316824, 26: A020985 and A020987, 27: A316825, 28: A159689, 29: A049320, 30: A003849, 31: A316826, 32: A316827, 33: A316828, 34: A316344, 35: A043529, 36: A316829, 37: A010060.
Cf. A001622, A104326, A132338.
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

Programs

  • Haskell
    a003849 n = a003849_list !! n
a003849_list = tail $ concat fws where
   fws = [1] : [0] : (zipWith (++) fws $ tail fws)
-- Reinhard Zumkeller, Nov 01 2013, Apr 07 2012
  • Magma
    t1:=[ n le 2 select ["0","0,1"][n] else Self(n-1) cat "," cat Self(n-2) : n in [1..12]]; t1[12];
  • Maple
    z := proc(m) option remember; if m=0 then [0] elif m=1 then [0,1] else [op(z(m-1)),op(z(m-2))]; fi; end; z(12);
M:=19; S[0]:=`0`; S[1]:=`01`; for n from 2 to M do S[n]:=cat(S[n-1], S[n-2]); od:
t0:=S[M]: l:=length(t0); for i from 1 to l do lprint(i-1,substring(t0,i..i)); od: # N. J. A. Sloane, Nov 01 2006
  • Mathematica
    Nest[ Flatten[ # /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 10] (* Robert G. Wilson v, Mar 05 2005 *)
Flatten[Nest[{#, #[[1]]} &, {0, 1}, 9]] (* IWABUCHI Yu(u)ki, Oct 23 2013 *)
Table[Floor[(n + 2) #] - Floor[(n + 1) #], {n, 0, 120}] &[2 - GoldenRatio] (* Michael De Vlieger, Aug 15 2016 *)
SubstitutionSystem[{0->{0,1},1->{0}},{0},{10}][[1]] (* Harvey P. Dale, Dec 20 2021 *)
  • PARI
    a(n)=my(k=2);while(fibonacci(k)<=n,k++);while(n>1,while(fibonacci(k--)>n,); n-=fibonacci(k)); n==1 \\ Charles R Greathouse IV, Feb 03 2014
  • PARI
    M3849=[2,2,1,0]/*L(k),S(k),L(k-1),S(k-1)*/; A003849(n)={while(n>M3849[1],M3849=vecextract(M3849,[1,2,1,2])+[M3849[3],M3849[4]<M. F. Hasler, Apr 07 2021
  • Python
    def fib(n):
    """Return the concatenation of A003849(0..F-1) where F is the smallest
       Fibonacci number > n, so that the result contains a(n) at index n."""
    a, b = '10'
    while len(b)<=n:
        a, b = b, b + a
    return b # Robert FERREOL, Apr 15 2016, edited by M. F. Hasler, Apr 07 2021
  • Python
    from math import isqrt
def A003849(n): return 2-(n+2+isqrt(m:=5*(n+2)**2)>>1)+(n+1+isqrt(m-10*n-15)>>1) # Chai Wah Wu, Aug 25 2022

Formula

a(n) = floor((n+2)*r) - floor((n+1)*r) where r=phi/(1+2*phi) and phi is the Golden Ratio. - Benoit Cloitre, Nov 10 2003
a(n) = A003714(n) mod 2 = A014417(n) mod 2. - Philippe Deléham, Jan 04 2004
The first formula by Cloitre is just one of an infinite family of formulas. Using phi^2=1+phi, it follows that r=phi/(1+2*phi)=2-phi. Then from floor(-x)=-floor(x)-1 for non-integer x, it follows that a(n)=2-A014675(n)=2-(floor((n+2)* phi)-floor((n+1)*phi)). - Michel Dekking, Aug 27 2016
a(n) = 1 - A096270(n+1), i.e., A096270 is the complement of this sequence. - A.H.M. Smeets, Mar 31 2024

Extensions

Revised by N. J. A. Sloane, Jul 03 2012

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Examples

			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, ...

References

  • 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.

Links

Crossrefs

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

Programs

  • Haskell
    a003842 n = a003842_list !! n
a003842_list = tail $ concat fws where
   fws = [2] : [1] : (zipWith (++) fws $ tail fws)
-- Reinhard Zumkeller, Oct 26 2013
  • Mathematica
    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 *)
  • PARI
    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
  • Python
    def A003842(length):
    a = [1]
    while len(a)Nicholas Stefan Georgescu, Jun 14 2022
  • Python
    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
  • Python
    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

Formula

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]

Extensions

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

A106750 Define the "Fibonacci" morphism f: 1->12, 2->1 and let a(0) = 2; then a(n+1) = f(a(n)).

Original entry on oeis.org

2, 1, 12, 121, 12112, 12112121, 1211212112112, 121121211211212112121, 1211212112112121121211211212112112, 1211212112112121121211211212112112121121211211212112121
Offset: 0

Views

Author

N. J. A. Sloane, May 16 2005. Initial term 2 added by N. J. A. Sloane, Jul 05 2012

Keywords

Comments

a(n) converges to the Fibonacci word A003842.
a(n) has length Fibonacci(n+1) (cf. A000045).

References

  • Berstel, Jean. "Fibonacci words—a survey." In The book of L, pp. 13-27. Springer Berlin Heidelberg, 1986.
  • 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.

Links

Crossrefs

Cf. A106748, A106749, A003842, A000045, A213975, A213976.

Programs

  • Mathematica
    FromDigits /@ NestList[ Flatten[ # /. {1 -> {1, 2}, 2 -> 1}] &, {2}, 8] (* Robert G. Wilson v, May 17 2005 *)

Extensions

More terms from Robert G. Wilson v, May 17 2005

A214208 First differences of A214207.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 2, 1, 5, 2, 1, 4, 1, 10, 1, 4, 1, 8, 2, 21, 1, 8, 2, 16, 1, 4, 41, 2, 16, 1, 4, 32, 1, 8, 83, 4, 32, 1, 8, 64, 2, 16, 1, 165, 8, 64, 2, 16, 1, 128, 4, 32, 1, 330, 16, 1, 128, 4, 32, 1, 256, 8, 64, 2, 661, 32, 1, 256, 8, 64, 2, 512, 16, 1, 128, 4, 1321, 64, 2, 512, 16, 1, 128, 4, 1024, 32, 1, 256, 8, 2642, 1, 128, 4, 1024, 32, 1, 256, 8
Offset: 1

Views

Author

N. J. A. Sloane, Jul 09 2012

Keywords

Comments

Created with the hope of obtaining a better understanding ofA213975.

Links

Crossrefs

Cf. A213975, A214207, A214209.

A214209 Numbers appearing in A214208 excluding powers 2^i with i>0.

Original entry on oeis.org

1, 3, 5, 10, 21, 41, 83, 165, 330, 661, 1321, 2642, 5285, 10569, 21139, 42277, 84554, 169109, 338217, 676435, 1352869, 2705738, 5411477, 10822953, 21645906, 43291813, 86583625, 173167251, 346334501, 692669002, 1385338005, 2770676009, 5541352018, 11082704037, 22165408073, 44330816147, 88661632293, 177323264586, 354646529173, 709293058345
Offset: 1

Views

Author

N. J. A. Sloane, Jul 09 2012

Keywords

Comments

Created with the hope of obtaining a better understanding of A213975.

Crossrefs

Cf. A213975, A214207, A214208.

Formula

It appears that a(n+1) = 2*a(n) + eps, where eps = 0, 1 or -1. The first 130 values of eps show a pattern which is somewhat mysterious.

A213976 a(n) = n-th term of A106750 reversed.

Original entry on oeis.org

2, 1, 21, 121, 21121, 12121121, 2112112121121, 121211212112112121121, 2112112121121121211212112112121121, 1212112121121121211212112112121121121211212112112121121, 21121121211211212112121121121211211212112121121121211212112112121121121211212112112121121
Offset: 0

Views

Author

N. J. A. Sloane, Jul 05 2012

Keywords

Comments

Arises in analyzing factors of A003842.

Links

Crossrefs

Cf. A003842, A106750, A213975.

A214216 List of minimal forbidden subwords of the Fibonacci word A003482.

Original entry on oeis.org

22, 111, 21212, 11211211, 2121121211212, 112112121121121211211, 2121121211211212112121121121211212, 1121121211211212112121121121211211212112121121121211211, 21211212112112121121211211212112112121121211211212112121121121211211212112121121121211212
Offset: 1

Views

Author

N. J. A. Sloane, Jul 10 2012

Keywords

Comments

If S is one of the terms of this sequence, then no word RS can appear as a subword of A003482.
Or, make a list of all words in {1,2}* that do not appear as factors of A003482 and discard any word which has a shorter word on the list as a right factor.
All the terms are palindromes.
Complementing the first and last digits of each term gives (essentially) A214217.

Examples

			A106750(3)=121 -> P=1 -> 111 = a(1).
A106750(4)=12112 -> P=121 -> 2121 = a(2).
A106750(5)=12112121 -> P=121121 -> 11211211 = a(3).

Links

Crossrefs

Cf. A003842, A106750, A213975, A214217.

Formula

To get a(n), take A106750(n+2), delete last two digits, producing a palindrome P, say. Then a(n) = 1P1 if n is odd, or 2P2 if n is even.
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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