Alexis Monnerot-Dumaine - cp's OEIS frontend

User: Alexis Monnerot-Dumaine

Alexis Monnerot-Dumaine has authored 9 sequences.

A308712 a(0) = 0 and a(1) = 1; for n > 1, a(n) = a(n-1)/2 if that number is an integer and not already in the sequence, otherwise a(n) = 3*a(n-1) + remainder of a(n-1)/2. (A variant of the Collatz sequence).

0, 1, 4, 2, 6, 3, 10, 5, 16, 8, 24, 12, 36, 18, 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 60, 30, 15, 46, 23, 70, 35, 106, 53, 160, 80, 240, 120, 360, 180, 90, 45, 136, 68, 204, 102, 51, 154, 77, 232, 116, 58, 29, 88, 44, 132, 66, 33, 100, 50, 25, 76, 38, 19, 58

Offset: 0

Alexis Monnerot-Dumaine, Jun 19 2019

Similar to A128333 and related to the 3x+1 (Collatz) sequence. Hits all positive integers?

			a(1)=1 => a(2)=3*1+1=4 because a(1) is odd => a(3)=4/2=2 because a(2) is even => a(4)=3*2+0=6 because a(3) is even but a(3)/2 is already in the sequence.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A126038, A005132, A128333.

a[0] = 0; a[1] = 1; a[n_] := a[n] = With[{b = a[n-1]}, If[EvenQ[b] && FreeQ[Array[a, n, 0], b/2], b/2, 3 b + Mod[b, 2]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 20 2019 *)

A171587 Sequence of the diagonal variant of the Fibonacci word fractal. Sequence of the Fibonacci tile.

Original entry on oeis.org

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

Offset: 0

Alexis Monnerot-Dumaine, Dec 12 2009

nonn

This is the upper Wythoff sequence (A001950) read mod 2 (for proof see formula section). So a(n) = floor((n+1)*phi^2) mod 2 where phi = (1+sqrt(5))/2. - Michel Dekking, Feb 01 2021
Interpreted as 0=turn right and 1=turn left, this sequence builds the diagonal variant of the Fibonacci word fractal. Base for the construction of the Fibonacci tile (Tiles the plane by translation in 2 ways).
From Michel Dekking, May 03 2018: (Start)
This is a morphic sequence, i.e., the letter to letter projection of a fixed point of a morphism. To see this, one uses the formula which generates (a(n)) from the Dense Fibonacci word A143667. Note that in the Dense Fibonacci word, which is the fixed point of the morphism
    0->10221, 1->1022, 2->1021,
the letter 0 exclusively occurs preceded directly by the letter 1. This enables one to create a new letter 3, encoding the word 10, and a morphism
    1->322, 2->321, 3->3223221,
which has the property that the letter to letter projection
    1->0, 2->1, 3->0
of its fixed point 3,2,2,3,2,2,1,3,2,1,... is equal to (a(n)).
(End)
Also Hofstadter G-sequence (A005206) mod 2. Another morphism can be written in octonary notation as: 0->4, 1->7, 2->5, 3->6, 4->60, 5->53, 6->71, 7->42, where the high bit gives A005614 and the low bit (i.e. "mod 2") gives this A171587 for n>0. The "Missing Words Proof Certificate" found under Links uses this representation to compute missing words of length L = 3, 4, 6, 9, 14, 22. Is there another missing word of length L = 35 as A001611 suggests? - Bradley Klee, Dec 24 2024

			q[2] = q[1]q[0] = 0,        q[3] = q[2]bar{q[1]} = 01,
q[4] = q[3]bar{q[2]} = 011, q[5] = q[4]q[3] = 01101.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
A. Blondin-Massé, S. Brlek, A. Garon, S. Labbé, Christoffel and Fibonacci Tiles, DGCI 2009. Lecture Notes in Computer Science, vol 5810.
A. Blondin-Massé, S. Brlek, A. Garon, S. Labbé, Christoffel and Fibonacci tiles, Sept 2009.
A. Blondin-Massé, S. Brlek, A. Garon, S. Labbé, Christoffel and Fibonacci tiles presentation, Sept 2009.
Bradley Klee, Missing Words Proof Certificate (Golang)
Bradley Klee, Turns Equivalence Proof Certificate (Golang)
Bradley Klee, Drawing of Fibonacci word fractal
Alexis Monnerot-Dumaine, The Fibonacci word fractal, Feb 2009.
Alexis Monnerot-Dumaine, The Fibonacci word fractal [Cached copy, with permission]

Cf. A143667, A003849. Cf. A379184, A379274, A379275.

Cf. A001950 (upper Wythoff sequence), A085002 (lower Wythoff sequence mod 2).

func b(n int) []int {
    a := make([]int, n+1);
    for i:=1; i < n+1; i++ {
        a[i] = i-a[a[i-1]];
    };
    for i:=0; i < n+1; i++ {
        a[i] %= 2;
    };
    return a
} // Bradley Klee, Dec 25 2024

(* This program supports the conjecture that A171587=(A001950 mod 2). *)
t = Nest[Flatten[# /. {1 -> {1, 0, 2, 2}, 0 -> {1, 0, 2, 2, 1}, 2 -> {1, 0, 2, 1}}] &, {1}, 5]
w = DeleteCases[t, 0] /. {1 -> 0, 2 -> 1}
u = Table[n + Floor[n*GoldenRatio], {n, 1, 500}]; v = Mod[u, 2]
Table[w[[n]] - v[[n]], {n, 1, 500}] (* supports conjecture for n=1,2,...,500 *)
(* t=A143667, w=A171587, u=A001950, conjecture: v=w *)

This sequence is defined by Blondin-Massé et al. as a limit of recursively defined words q[n]. Here q[0] is the empty word, and q[1]=0.
The recursion is given by
    q[n]=q[n-1]q[n-2] if n=2 mod 3, and
    q[n]=q[n-1]bar{q[n-2]} if n=0 or 1 mod 3,
where bar exchanges 0 and 1.
Also application of the mapping 1->0, 2->1, 0->empty word to the Dense Fibonacci word A143667.
Conjecture: A171587=(A001950 mod 2), as suggested for n=1,2,...,500 by Mathematica program below. - Clark Kimberling, May 31 2011
From Michel Dekking, May 03 2018: (Start)
Proof of Kimberling's 2011 conjecture, i.e., this sequence is the parity sequence of the Upper Wythoff sequence A001950.
The first difference sequence 3, 2, 3, 3, 2, 3, 2, 3, ... of the Upper Wythoff sequence is equal to the unique fixed point of the morphism
  beta:  2 -> 3, 3 -> 32 (cf. A282162).
We define the first difference operator D on finite words w by
    D(w(1)...w(m)) = (w(2)-w(1))...(w(m)-w(m-1)).
Note that the length of D(w) is one less than the length of w, and note
LEMMA 1: D(vw) = D(v)|w(1)-v(l)|D(w), if v = v(1)...v(l), and w = w(1)...w(m). Here |w(1)-v(l)| is modulo 2.
We also need (easily proved by induction)
LEMMA 2: The last letter of the word q[n] equals 0 if and only if n = 0,1,2 modulo 6.
Almost trivial is
LEMMA 3: The last letter e(n) of beta^n(2) equals 2 if and only if n = 0 modulo 2.
The following proposition implies the conjecture.
PROPOSITION: The difference sequence of q[n] satisfies D(q[n]) = beta^{n-1}(2) e(n-1)^{-1} modulo 2 for n>3.
Note that, by definition, beta^n(2) e(n)^{-1} is just the word beta^n(2), with the last letter removed.
PROOF: By induction. Combine Lemma 1, 2 and 3 in the recursion for the q[n], for n = 0,...,5 modulo 6, using the following table:
n modulo 6               | 0 | 1 | 2 | 3 | 4 | 5 |
last letter of q[n-1]    | 1 | 0 | 0 | 0 | 1 | 1 |
first letter of q[n-2]*  | 1 | 1 | 0 | 1 | 1 | 0 |
Here q[n-2]* denotes either q[n-2] (if n == 2 (mod 3)), or bar{q[n-2]} (if n == 0,1 (mod 3)).
For example, where all equalities are modulo 2,
    D(q[8]) = D(q[7]) 0 D(q[6]) = beta^6(2) f(6) 0 beta^5(2) f(5) = beta^6(2) beta^5(2) f(5) = beta^5(32) f(5) = beta^7(2) f(7),
where f(n):=(e(n) mod 2)^{-1}.
(End)

Formula corrected and extended by Michel Dekking, May 03 2018

A171990 Least integer a(n) for which the iterated function log, iterated n times, is defined.

Original entry on oeis.org

1, 2, 3, 16, 3814280

Offset: 1

Alexis Monnerot-Dumaine, Jan 21 2010

nonn

Log(a(1)) is defined if a(1) > 0, so a(1) = 1.
Log(log(a(2))) is defined if log(a(2)) > 0 => a(2) > 1 => a(2) = 2.
The sequence grows rapidly: a(6) = 2.33150...10^1656520, and is too large to include here.

			a(2) = 2 because log(log(2)) is defined and log(log(1)) is not;
a(3) = 3 because log(log(log(3))) is defined;
a(4) = 16 because log(log(log(log(16)))) is defined.
From _Robert G. Wilson v_, Jul 05 2022: (Start)
a(3) = ceiling(A001113).
a(4) = ceiling(A073226).
a(5) = ceiling(A073227).
a(6) = ceiling(A085667). (End)

Cf. A074785 (log(log(2))), A085667, A004002, A001113, A073226, A073227, A085667.

a(n) = my(k=1); while(1, my(s=k, i=0); while(s > 0, s=log(s); if(s > 0, i++)); if(i==n-1, return(k)); k++) \\ Felix Fröhlich, Nov 22 2015

For n > 2, a(n) = ceiling(e^(e^(...))) where e appears n-2 times.

A171588 The Pell word: Fixed point of the morphism 0->001, 1->0.

Original entry on oeis.org

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

Offset: 1

Alexis Monnerot-Dumaine, Dec 12 2009

From Peter Bala, Nov 22 2013: (Start)
This is a Sturmian word: equals the limit word S(infinity) where S(0) = 0, S(1) = 001 and for n >= 1, S(n+1) = S(n)S(n)S(n-1). See the examples below.
This sequence corresponds to the case k = 2 of the Sturmian word S_k(infinity) as defined in A080764. See A159684 for the case k = 1. (End)
Characteristic word with slope 1 - 1/sqrt(2). Since the characteristic word with slope 1-theta is the mirror image of the characteristic word with slope theta, a(n)= 1 - A080764(n) for all n. - Michel Dekking, Jan 31 2017
The positions of 0 comprise A001951 (Beatty sequence for sqrt(2)); the positions of 1 comprise A001952 (Beatty sequence for 2+sqrt(2)). - Clark Kimberling, May 11 2017
This is also the fixed point of the mapping 00->0010, 01->001, 10->010, starting with 00 [Dekking and Keane, 2022]. See A289001. - N. J. A. Sloane, Mar 09 2022

			From _Peter Bala_, Nov 22 2013: (Start)
The sequence of words S(n) begins
  S(0) = 0
  S(1) = 001
  S(2) = 001 001 0
  S(3) = 0010010 0010010 001
  S(4) = 00100100010010001 00100100010010001 0010010.
The lengths of the words are [1, 3, 7, 17, 41, ...] = A001333 (apart from the initial term).  (End)

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 284.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Scott Balchin and Dan Rust, Computations for Symbolic Substitutions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.1.
Jean Berstel and Juhani Karhumäki, Combinatorics on words-a tutorial. Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, 79:178-228, 2003.
Michel Dekking, Substitution invariant Sturmian words and binary trees, arXiv:1705.08607 [math.CO], 2017.
Michel Dekking, Substitution invariant Sturmian words and binary trees, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A17.
Michel Dekking and Mike Keane, Two-block substitutions and morphic words, arXiv:2202.13548 [math.CO], 2022.
M. Lothaire, Combinatorics on Words.
Wikipedia, Sturmian word

Cf. A000129, A001333, A001951, A001952, A003849, A080764, A159684, A289001.

[Floor((n+1)*(1-1/Sqrt(2))-Floor(n*(1-1/Sqrt(2)))): n in [1..100]]; // Vincenzo Librandi, Jan 31 2017

Digits := 50: u := evalf(2 + sqrt(2)): A171588 := n->floor((n+1)/u) - floor(n/u): seq(A171588(n), n = 1..80); # Peter Bala, Nov 22 2013

Table[Floor[(n + 1) (1 - 1/Sqrt[2]) - Floor[n (1 - 1/Sqrt[2])]], {n, 100}] (* Vincenzo Librandi, Jan 31 2017 *)
Nest[Flatten[# /. {0 -> {0, 0, 1}, 1 -> {0}}] &, {0}, 6] (* Clark Kimberling, May 11 2017 *)

from math import isqrt
def A171588(n): return 1+isqrt(n**2>>1)-isqrt((n+1)**2>>1) # Chai Wah Wu, May 24 2025

a(n) = floor((n + 1)/(2 + sqrt(2))) - floor(n /(2 + sqrt(2))). - Peter Bala, Nov 22 2013

a(n) = floor((n+1)*(1 - 1/sqrt(2))) - floor(n*(1 - 1/sqrt(2))). - Michel Dekking, Jan 31 2017

A156596 Infinite Fibonacci word fractal sequence.

Original entry on oeis.org

1, 0, 1, 2, 0, 2, 0, 2, 1, 0, 1, 2, 0, 2, 0, 2, 1, 0, 1, 0, 1, 2, 0, 2, 1, 0, 1, 0, 1, 2, 0, 2, 1, 0, 1, 0, 1, 2, 0, 2, 0, 2, 1, 0, 1, 2, 0, 2, 0, 2, 1, 0, 1, 0, 1, 2, 0, 2, 1, 0, 1, 0, 1, 2, 0, 2, 1, 0, 1, 0, 1, 2, 0, 2, 0, 2, 1, 0, 1, 2, 0, 2, 0, 2, 1, 0, 1, 2, 0, 2, 0, 2, 1, 0, 1, 0, 1, 2, 0, 2, 1, 0, 1, 0, 1

Offset: 1

Alexis Monnerot-Dumaine, Feb 10 2009

Apply to A143667 the map : 0 -> 12, 1 -> 10, 2 -> 02. or apply to A003849 (the Fibonacci word), after grouping the terms 2 by 2, the map : "00" -> "12", "01"->"10, "10"->"02". Draws the Fibonacci word fractal curve when applying the following drawing rule: if "0" then draw a segment forward, if "1" then draw a segment forward and turn 90A degs right, if "2" the draw segment and turn 90A degs left.

M. Lothaire, Combinatorics on words, Cambridge University Press.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
A. Monnerot-Dumaine, Fibonacci word fractal
Alexis Monnerot-Dumaine, The Fibonacci word fractal [Cached copy, with permission]

Cf. A003849, A143667.

a143667 n = a143667_list !! (n-1)
a143667_list = f a003849_list where
   f (0:0:ws) = 0 : f ws; f (0:1:ws) = 1 : f ws; f (1:0:ws) = 2 : f ws
-- Reinhard Zumkeller, Jul 29 2014

Partition[Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}]&, {0}, 10], 2] /. {{0, 0} -> {1, 2}, {0, 1} -> {1, 0}, {1, 0} -> {0, 2}} // Flatten (* Jean-François Alcover, Jul 16 2015 *)

A156595 Fixed point of the morphism 0->011, 1->010.

Original entry on oeis.org

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

Offset: 0

Alexis Monnerot-Dumaine, Feb 10 2009

Start with 0 and apply the morphism 0->011 and 1->010 repeatedly.
This sequence draws the Sierpinski gasket, when iterating the following odd-even drawing rule: If "1" then draw a segment forward, if "0" then draw a segment forward and turn 120 degrees right if in odd position or left if in even position.
From Dimitri Hendriks, Jun 29 2010: (Start)
This sequence is the first difference of the Mephisto Waltz A064990, i.e., a(n) = A064990(n) + A064990(n+1), where '+' is addition modulo 2.
This sequence can also be generated as a Toeplitz word: First consider the periodic word 0,1,$,0,1,$,0,1,$,... and then fill the gaps $ by the bitwise negation of the sequence itself: 0,1,1,0,1,0,0,1,0,.... See the Allouche/Bacher reference for a precise definition of Toeplitz sequences. (End)
From Joerg Arndt, Jan 21 2013: (Start)
Identical to the morphism 0-> 011010010, 1->011010011 given on p. 100 of the Fxtbook (see link), because 0 -> 011 -> 011010010 and 1 -> 010 -> 011010011.
This sequence gives the turns (by 120 degrees) of the R9-dragon curve (displayed on p. 101) which can be rendered as follows:
  [Init] Set n=0 and direction=0.
  [Draw] Draw a unit line (in the current direction). Turn left/right if a(n) is zero/nonzero respectively.
  [Next] Set n=n+1 and goto (draw).
(End)

			0 -> 0,1,1 -> 0,1,1,0,1,0,0,1,0 -> ...

M. Lothaire, Combinatorics on words.

J.-P. Allouche and R. Bacher, Toeplitz Sequences, Paperfolding, Towers of Hanoi, and Progression-Free Sequences of Integers, L'Enseignement Mathématique, volume 38, pages 315-327, 1992.
Joerg Arndt, Matters Computational (The Fxtbook) (section 1.31.5 "Dragon curves based on radix-R counting", pp. 95-101, image on p. 101).
Gabriele Fici and Jeffrey Shallit, Properties of a Class of Toeplitz Words, arXiv:2112.12125 [cs.FL], 2021.
Kevin Ryde, Iterations of the Terdragon Curve, see index "AltTurnRpred" with AltTurnRpred(n) = a(n-1).
Index entries for sequences that are fixed points of mappings

Cf. A278996 (indices of 0's), A278997 (indices of 1's), A189717 (partial sums).
Cf. A189628 (morphisms guide).
Cf. A307672 (draws curves that align with the Sierpinski gasket).

Nest[ Flatten[ # /. {0 -> {0, 1, 1}, 1 -> {0, 1, 0}}] &, {0}, 10]
SubstitutionSystem[{0->{0,1,1},1->{0,1,0}},0,{5}][[1]] (* Harvey P. Dale, Jan 15 2022 *)

from sympy import integer_log
def A156595(n): return sum(((m:=(n+1)//9**i)-2)//9+(m-3)//9+(m-5)//9+(m-8)//9+4 for i in range(integer_log(n+1,9)[0]+1))-sum(((m:=n//9**i)-2)//9+(m-3)//9+(m-5)//9+(m-8)//9+4 for i in range(integer_log(n,9)[0]+1)) if n else 0 # Chai Wah Wu, Feb 16 2025

a(3k-2)=0, a(3k-1)=1, a(3k)=1-a(k) for k>=1, a(0)=0. - Clark Kimberling, Apr 28 2011

A143667 Digits of the infinite Fibonacci word A003849 grouped 2 by 2 and interpreted as a binary value.

Original entry on oeis.org

1, 0, 2, 2, 1, 0, 2, 2, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 0, 2, 2, 1, 0, 2, 2, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 0, 2, 2, 1, 0, 2, 2, 1, 0, 2, 2, 1, 1, 0, 2, 1, 1, 0, 2, 2, 1, 0, 2, 2, 1, 0, 2, 2, 1, 1, 0, 2, 1, 1, 0, 2, 2, 1, 0, 2, 2, 1, 0, 2, 2, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 0, 2, 2, 1, 0, 2, 2, 1, 1, 0

Offset: 1

Alexis Monnerot-Dumaine, Aug 28 2008

Group 2 by 2 the successive letters of the infinite Fibonacci word A003849 then apply: 00->0, 01->1 and 10->2.
Also result of the following iterated morphism: 1->1022, 0->10221, 2->1021, iterated from letter 1. (Monnerot 2008)
Fractal properties studied (proposed for publication)
(a(n)) is essentially the same sequence as A123564. Simply change the alphabet to {1,2,3}, and permute the letters. The Standard Form of (a(n)) written as a word on the alphabet {a,b,c} is abccabccaabc... . Other forms for this standard form are 1,2,3,3,1,2,3,3,1,1,2,3,.... and 123312331123... - _Michel Dekking, Oct 07 2017
(a(n)) is the fixed point of the 2-block map (called 2-block Fibonacci to the power 3) 00->0100101001, 01->01001010, 10->01001001, followed by the coding above. - Michel Dekking, Sep 26 2017

			a(1) = 1 because the infinite Fibonacci word starts with "01", a(2) = 0 because it continues with "00", and so on.

M. Lothaire, Combinatorics on words, Cambridge University Press.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
J.-P. Allouche, M. Mendès France, and G. Skordev, Non-intersectivity of Paperfolding Dragon Curves and of Curves Generated by Automatic Sequences, INTEGERS, Electronic Journal of Combinatorial Number Theory, vol. 18A, Article #A2, 2018. Mentions this sequence.
Wieb Bosma and Henk Don, Constructing Morphisms for Arithmetic Subsequences of Fibonacci, Ch. 6, Logics and Type Systems in Theory and Practice (2024) Lect. Notes Comp. Sci. (LNCS) Vol. 14560, 100-110.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
Michel Dekking and Mike Keane, Two-block substitutions and morphic words, arXiv:2202.13548 [math.CO], 2022.
A. Monnerot-Dumaine, The Fibonacci Word Fractal.
Alexis Monnerot-Dumaine, The Fibonacci word fractal [Cached copy, with permission]
J. L. Ramírez and G. N. Rubiano, Properties and Generalizations of the Fibonacci Word Fractal, The Mathematica Journal, Vol. 16 (2014). See "Dense Fibonacci word". - _N. J. A. Sloane_, Mar 26 2014

Cf. A003849, A123564.

a143667 n = a143667_list !! (n-1)
a143667_list = f a003849_list where
   f (0:0:ws) = 0 : f ws; f (0:1:ws) = 1 : f ws; f (1:0:ws) = 2 : f ws
-- Reinhard Zumkeller, Jul 29 2014

Table[3 - (Floor[#1 #2] - 2 Floor[#1 (#2 - 1)] + Floor[#1 (#2 + 1)]) & @@ {1/GoldenRatio, 2 n}, {n, 100}] (* Michael De Vlieger, Oct 06 2017 *)

a(n) = decimal value of b(2n-1)b(2n), b(n) taken from A003849 (infinite Fibonacci word).

A143668 Result of the morphing 01->01021212, 02->0102121201, 12->01021201, iterated from '01'. Sequence of the Fibonacci word fractal.

Original entry on oeis.org

0, 1, 0, 2, 1, 2, 1, 2, 0, 1, 0, 2, 1, 2, 1, 2, 0, 1, 0, 1, 0, 2, 1, 2, 0, 1, 0, 1, 0, 2, 1, 2, 0, 1, 0, 1, 0, 2, 1, 2, 1, 2, 0, 1, 0, 2, 1, 2, 1, 2, 0, 1, 0, 1, 0, 2, 1, 2, 0, 1, 0, 1, 0, 2, 1, 2, 0, 1, 0, 1, 0, 2, 1, 2, 1, 2, 0, 1, 0, 2, 1, 2, 1, 2, 0, 1, 0, 2, 1, 2, 1, 2, 0, 1, 0, 1, 0, 2, 1, 2

Offset: 1

Alexis Monnerot-Dumaine, Aug 28 2008

nonn

Letter '2' is always in an even position and '0' an odd position.
When replacing '2' by '0', equals the infinite Fibonacci word (see A003849).
This sequence produces the Fibonacci word fractal when applying the following turtle graphics rules: 0->draw segment+turn right, 1-> draw segment, 2-> draw segment+turn left (A. Monnerot-Dumaine 2008 see links).
This sequence is the [1->12, 2->01, 3->02]-transform of A123564. - Michel Dekking, Mar 03 2018

M. Lothaire, Combinatorics on words, Cambridge University press.

Alexis Monnerot-Dumaine, The Fibonacci Word Fractal, HAL Id : hal-00367972, 2009.
Alexis Monnerot-Dumaine, The Fibonacci word fractal [Cached copy, with permission]

Cf. A003849, A123564.

Let (b(n)) be the infinite Fibonacci word. if (b(n)=0 and n is even), then a(n)=2, else a(n)=b(n).

A120384 Isolated primes: geometric mean of distances of a prime to neighboring primes sets record.

Original entry on oeis.org

3, 5, 7, 23, 53, 89, 113, 211, 1259, 1327, 1847, 2179, 2503, 5623, 14107, 19661, 24281, 38501, 58831, 268343, 396833, 1272749, 2198981, 3863107, 4411963, 4958131, 5102953, 7950001, 8917523, 10938023, 12623189, 22440841, 24662467, 32616223

Offset: 1

Alexis Monnerot-Dumaine, Jun 29 2006

nonn

A096265 is based on arithmetic mean or total distance to neighbors. But it doesn't say if it is isolated from them or close to one of them.

			a(4) = 23 because the distance (geometric mean) to its neighbors (19 and 29) equals = sqrt(4*6) = 4.8989. No smaller prime is more isolated. The next more isolated prime a(5) is 53.

Ken Takusagawa, Table of n, a(n) for n = 1..54

Cf. A096265.

lista(nn) = {d = 0; p = 1; q = 2; r = 3; for (i=1, nn, p = q; q = r; r = nextprime(r+1); if ((nd = (q-p)*(r-q)) > d, print1(q, ", "); d = nd;););} \\ Michel Marcus, Jun 12 2013

Offset corrected and a(22)-a(34) added by Donovan Johnson, May 23 2010

cp's OEIS Frontend

User: Alexis Monnerot-Dumaine

A308712 a(0) = 0 and a(1) = 1; for n > 1, a(n) = a(n-1)/2 if that number is an integer and not already in the sequence, otherwise a(n) = 3*a(n-1) + remainder of a(n-1)/2. (A variant of the Collatz sequence).

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A171587 Sequence of the diagonal variant of the Fibonacci word fractal. Sequence of the Fibonacci tile.

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A171990 Least integer a(n) for which the iterated function log, iterated n times, is defined.

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A171588 The Pell word: Fixed point of the morphism 0->001, 1->0.

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A156596 Infinite Fibonacci word fractal sequence.

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A156595 Fixed point of the morphism 0->011, 1->010.

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A143667 Digits of the infinite Fibonacci word A003849 grouped 2 by 2 and interpreted as a binary value.

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