A276800 -id:A276800 - cp's OEIS frontend

A058265 Decimal expansion of the tribonacci constant t, the real root of x^3 - x^2 - x - 1.

1, 8, 3, 9, 2, 8, 6, 7, 5, 5, 2, 1, 4, 1, 6, 1, 1, 3, 2, 5, 5, 1, 8, 5, 2, 5, 6, 4, 6, 5, 3, 2, 8, 6, 6, 0, 0, 4, 2, 4, 1, 7, 8, 7, 4, 6, 0, 9, 7, 5, 9, 2, 2, 4, 6, 7, 7, 8, 7, 5, 8, 6, 3, 9, 4, 0, 4, 2, 0, 3, 2, 2, 2, 0, 8, 1, 9, 6, 6, 4, 2, 5, 7, 3, 8, 4, 3, 5, 4, 1, 9, 4, 2, 8, 3, 0, 7, 0, 1, 4

Offset: 1

Robert G. Wilson v, Dec 07 2000

"The tribonacci constant, the only real solution to the equation x^3 - x^2 - x - 1 = 0, which is related to tribonacci sequences (in which U_n = U_n-1 + U_n-2 + U_n-3) as the Golden Ratio is related to the Fibonacci sequence and its generalizations. This ratio also appears when a snub cube is inscribed in an octahedron or a cube, by analogy once again with the appearance of the Golden Ratio when an icosahedron is inscribed in an octahedron. [John Sharp, 1997]"
The tribonacci constant corresponds to the Golden Section in a tripartite division 1 = u_1 + u_2 + u_3 of a unit line segment; i.e., if 1/u_1 = u_1/u_2 = u_2/u_3 = c, c is the tribonacci constant. - Seppo Mustonen, Apr 19 2005
The other two polynomial roots are the complex-conjugated pair -0.4196433776070805662759262... +- i* 0.60629072920719936925934... - R. J. Mathar, Oct 25 2008
For n >= 3, round(q^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. - Vladimir Shevelev, Mar 21 2014
Concerning orthogonal projections, the tribonacci constant is the ratio of the diagonal of a square to the width of a rhombus projected by rotating a square along its diagonal in 3D until the angle of rotation equals the apparent apex angle at approximately 57.065 degrees (also the corresponding angle in the formula generating A256099). See illustration in the links. - Peter M. Chema, Jan 02 2017
From Wolfdieter Lang, Aug 10 2018: (Start)
Real eigenvalue t of the tribonacci Q-matrix <<1, 1, 1>,<1, 0, 0>,<0, 1, 0>>.
Limit_{n -> oo} T(n+1)/T(n) = t (from the T recurrence), where T = {A000073(n+2)}_{n >= 0}. (End)
The nonnegative powers of t are t^n = T(n)*t^2 + (T(n-1) + T(n-2))*t + T(n-1)*1, for n >= 0, with T(n) = A000073(n), with T(-1) = 1 and T(-2) = -1, This follows from the recurrences derived from t^3 = t^2 + t + 1. See the examples below. For the negative powers see A319200. - Wolfdieter Lang, Oct 23 2018
Note that we have: t + t^(-3) = 2, and the k-nacci constant approaches 2 when k approaches infinity (Martin Gardner). - Bernard Schott, May 16 2022
The roots of this cubic are found from those of y^3 - (4/3)*y - 38/27, after adding 1/3. - Wolfdieter Lang, Aug 24 2022
The algebraic number t - 1 has minimal polynomial x^3 + 2*x^2 - 2 over Q. The roots coincide with those of y^3 - (4/3)*y - 38/27, after subtracting 2/3. - Wolfdieter Lang, Sep 20 2022
The value of the ratio R/r of the radius R of a uniform ball to the radius r of a spherical hole in it with a common point of contact, such that the center of gravity of the object lies on the surface of the spherical hole (Schmidt, 2002). - Amiram Eldar, May 20 2023

			1.8392867552141611325518525646532866004241787460975922467787586394042032220\
    81966425738435419428307014141979826859240974164178450746507436943831545\
    820499513796249655539644613666121540277972678118941041...
From _Wolfdieter Lang_, Oct 23 2018: (Start)
The coefficients of t^2, t, 1 for t^n begin, for n >= 0:
    n     t^2    t    1
    -------------------
    0      0     0    1
    1      0     1    0
    2      1     0    0
    1      1     1    1
    4      2     2    1
    5      4     3    2
    6      7     6    4
    7     13    11    7
    8     24    20   13
    9     44    37   24
   10     81    68   44
...  (End)

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.2.
Martin Gardner, The Second Scientific American Book of Mathematical Puzzles and Diversions, "Phi: The Golden Ratio", Chapter 8, p. 101, Simon & Schuster, NY, 1961.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, page 23.

Harry J. Smith, Table of n, a(n) for n = 1..20000
A. Beha et al., The convergence of diffy boxes, American Mathematical Monthly, Vol. 112 (2005), pp. 426-439.
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), Article P1.52.
O. Deveci, Y. Akuzum, E. Karaduman, and O. Erdag, The Cyclic Groups via Bezout Matrices, Journal of Mathematics Research, Vol. 7, No. 2 (2015), pp. 34-41.
Ömür Deveci, Zafer Adıgüzel, and Taha Doğan, On the Generalized Fibonacci-circulant-Hurwitz numbers, Notes on Number Theory and Discrete Mathematics, Vol. 26, No. 1 (2020), 179-190.
Peter M. Chema, Tribonacci constant as ratio of square to rhombus projection.
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
S. Litsyn and Vladimir Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, Vol. 1, No. 4 (2005), 499-512.
Xerardo Neira, A geometric construction of the tribonacci constant with marked ruler and compass.
Tito Piezas III, Tribonacci constant and Pi.
Simon Plouffe, Tribonacci constant to 2000 digits.
Simon Plouffe, The Tribonacci constant(to 1000 digits).
Herbert C. H. Schmidt, Problem 2670, Crux Mathematicorum, Vol. 28, No. 7 (2002), pp. 464-465.
Vladimir Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014).
Nikita Sidorov, Expansions in non-integer bases: Lower, middle and top orders, Journal of Number Theory, Volume 129, Issue 4, April 2009, Pages 741-754. See Lemma 4.1 p. 750.
Kees van Prooijen, The Odd Golden Section.
Kees van Prooijen, Tribonacci Box (analog of Golden Rectangle).
Eric Weisstein's World of Mathematics, Tribonacci Number.
Eric Weisstein's World of Mathematics, Tribonacci Constant.
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.
Index entries for algebraic numbers, degree 3

Cf. A000073, A019712 (continued fraction), A133400, A254231, A158919 (spectrum = floor(n*t)), A357101 (x^3-2*x^2-2).
Cf. A192918 (reciprocal), A276800 (square), A276801 (cube), A319200.
k-nacci constants: A001622 (Fibonacci), this sequence (tribonacci), A086088 (tetranacci), A103814 (pentanacci), A118427 (hexanacci), A118428 (heptanacci).

Digits:=200; fsolve(x^3=x^2+x+1); # N. J. A. Sloane, Mar 16 2019

RealDigits[ N[ 1/3 + 1/3*(19 - 3*Sqrt[33])^(1/3) + 1/3*(19 + 3*Sqrt[33])^(1/3), 100]] [[1]]
RealDigits[Root[x^3-x^2-x-1,1],10,120][[1]] (* Harvey P. Dale, Mar 23 2019 *)

set_display(none)$ fpprec:100$ bfloat(rhs(solve(t^3-t^2-t-1,t)[3])); /* Dimitri Papadopoulos, Nov 09 2023 */

default(realprecision, 20080); x=solve(x=1, 2, x^3 - x^2 - x - 1); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b058265.txt", n, " ", d));  \\ Harry J. Smith, May 30 2009

q=(1+sqrtn(19+3*sqrt(33),3)+sqrtn(19-3*sqrt(33),3))/3 \\ Use \p# to set 'realprecision'. - M. F. Hasler, Mar 23 2014

t = (1/3)*(1+(19+3*sqrt(33))^(1/3)+(19-3*sqrt(33))^(1/3)). - Zak Seidov, Jun 08 2005
t = 1 - Sum_{k>=1} A057597(k+2)/(T_k*T_(k+1)), where T_n = A000073(n+1). - Vladimir Shevelev, Mar 02 2013
1/t + 1/t^2 + 1/t^3 = 1/A058265 + 1/A276800 + 1/A276801 = 1. - N. J. A. Sloane, Oct 28 2016
t = (4/3)*cosh((1/3)*arccosh(19/8)) + 1/3. - Wolfdieter Lang, Aug 24 2022
t = 2 - Sum_{k>=0} binomial(4*k + 2, k)/((k + 1)*2^(4*k + 3)). - Antonio Graciá Llorente, Oct 28 2024

A003145 Positions of letter b in the tribonacci word abacabaabacababac... generated by a->ab, b->ac, c->a (cf. A092782).

Original entry on oeis.org

2, 6, 9, 13, 15, 19, 22, 26, 30, 33, 37, 39, 43, 46, 50, 53, 57, 59, 63, 66, 70, 74, 77, 81, 83, 87, 90, 94, 96, 100, 103, 107, 111, 114, 118, 120, 124, 127, 131, 134, 138, 140, 144, 147, 151, 155, 158, 162, 164, 168, 171, 175, 179, 182, 186, 188, 192, 195, 199, 202, 206, 208

Offset: 1

N. J. A. Sloane

nonn

A003144, A003145, A003146 may be defined as follows. Consider the map psi: a -> ab, b -> ac, c -> a. The image (or trajectory) of a under repeated application of this map is the infinite word a, b, a, c, a, b, a, a, b, a, c, a, b, a, b, a, c, ... (setting a = 1, b = 2, c = 3 gives A092782). The indices of a, b, c give respectively A003144, A003145, A003146. - Philippe Deléham, Feb 27 2009
The infinite word may also be defined as the limit S_oo where S_1 = a, S_n = psi(S_{n-1}). Or, by S_1 = a, S_2 = ab, S_3 = abac, and thereafter S_n = S_{n-1} S_{n-2} S_{n-3}. It is the unique word such that S_oo = psi(S_oo).
Also indices of b in the sequence closed under a -> abac, b -> aba, c -> ab; starting with a(1) = a. - Philippe Deléham, Apr 16 2004
Theorem: A number m is in this sequence iff the tribonacci representation of m-1 ends with 01. [Duchene and Rigo, Remark 2.5] - N. J. A. Sloane, Mar 02 2019

Eric Duchêne, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, Urban Larsson, Wythoff Visions, Games of No Chance, Vol. 5; MSRI Publications, Vol. 70 (2017), pages 101-153.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..10609
Elena Barcucci, Luc Belanger and Srecko Brlek, On tribonacci sequences, Fib. Q., 42 (2004), 314-320.
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Fibonacci representations of higher order, Fib. Quart., 10 (1972), 43-69. The present sequence is called b.
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. doi:10.1051/ita:2007039. [Also available from Numdam]
A. J. Hildebrand, Junxian Li, Xiaomin Li, Yun Xie, Almost Beatty Partitions, arXiv:1809.08690 [math.NT], 2018.
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.

Cf. A003144, A003146, A080843, A092782, A058265, A276799, A276800, A276794, A276797.
First differences give A276789. A278040 (subtract 1 from each term, and use offset 1).
For tribonacci representations of numbers see A278038.

M:=17; S[1]:=`a`; S[2]:=`ab`; S[3]:=`abac`;
for n from 4 to M do S[n]:=cat(S[n-1], S[n-2], S[n-3]); od:
t0:=S[M]: l:=length(t0); t1:=[];
for i from 1 to l do if substring(t0,i..i) = `b` then t1:=[op(t1),i]; fi; od: # N. J. A. Sloane

StringPosition[SubstitutionSystem[{"a" -> "ab", "b" -> "ac", "c" -> "a"}, "b", {#}][[1]], "b"][[All, 1]] &@9 (* Michael De Vlieger, Mar 30 2017, Version 10.2, after JungHwan Min at A003144 *)

It appears that a(n) = floor(n*t^2) + eps for all n, where t is the tribonacci constant A058265 and eps is 0, 1, or 2. See A276799. - N. J. A. Sloane, Oct 28 2016. This is true - see the Dekking et al. paper. - N. J. A. Sloane, Jul 22 2019

More terms from Philippe Deléham, Apr 16 2004
Corrected by T. D. Noe and N. J. A. Sloane, Nov 01 2006
Entry revised by N. J. A. Sloane, Oct 13 2016

A140103 Term-by-term sums of A140101 and A140100; also, equals the complement of A140102, which is the term-by-term differences of A140101 and A140100, where A140101 is the complement of A140100.

Original entry on oeis.org

3, 8, 12, 17, 20, 25, 29, 34, 39, 43, 48, 51, 56, 60, 65, 69, 74, 77, 82, 86, 91, 96, 100, 105, 108, 113, 117, 122, 125, 130, 134, 139, 144, 148, 153, 156, 161, 165, 170, 174, 179, 182, 187, 191, 196, 201, 205, 210, 213, 218, 222, 227, 232, 236, 241, 244, 249

Offset: 1

Paul D. Hanna, Jun 04 2008

nonn

Conjecture: a(n) = A003145(n) + n. This is the most direct connection between the Greedy Queens sequence and the tribonacci word that I know. - Michel Dekking, Mar 19 2019. [My notes show that I made this conjecture on Jul 20 2018. There are many similar conjectures relating the two problems. For example A140100 = A003145(n)-A003144(n), A140101(n) = A003146(n)-A003145(n), a(n) = A003146(n)-A003144(n). - N. J. A. Sloane, Mar 19 2019] All these conjectures are now theorems - see the Dekking et al. paper. - N. J. A. Sloane, Jul 22 2019

Robbert Fokkink, Gerard Francis Ortega, Dan Rust, Corner the Empress, arXiv:2204.11805. See Table 2.

N. J. A. Sloane, Table of n, a(n) for n=1..50000, Sep 13 2016 (First 1001 terms from Reinhard Zumkeller)
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.
N. J. A. Sloane, Maple program for A140100, A140101, A140102, A140103

Cf. A140102 (complement); A140100, A140101; A058265, A276800.

For first differences of A140100, A140101, A140102, A140103 see A305392, A305374, A305393, A305394.

Maple
```
See link.
```

nmax = 100; y[0] = 0; x[1] = 1; y[1] = 2; x[n_] := x[n] = For[yn = y[n-1] + 1, True, yn++, For[xn = x[n-1] + 1, xn < yn, xn++, xx = Array[x, n-1]; yy = Array[y, n-1]; If[FreeQ[xx, xn | yn] && FreeQ[yy, xn | yn] && FreeQ[yy - xx, yn - xn] && FreeQ[yy + xx, yn - xn], y[n] = yn; Return[xn]]]];
Do[x[n], {n, 1, nmax}];
yy + xx (* Jean-François Alcover, Aug 01 2018 *)

{X=[1];Y=[2];D=[1];S=[3];print1(Y[1]-X[1]","); for(n=1,100,for(j=2,2*n,if(setsearch(Set(concat(X,Y)),j)==0,Xt=concat(X,j); for(k=j+1,3*n,if(setsearch(Set(concat(Xt,Y)),k)==0, if(setsearch(Set(concat(D,S)),k-j)==0,if(setsearch(Set(concat(D,S)),k+j)==0, X=Xt;Y=concat(Y,k);D=concat(D,k-j);S=concat(S,k+j); print1(Y[ #X]-X[ #Y]",");break);break))))))}

a(n) = A140100(n) + A140101(n).

Conjecture: the limit of A140103(n)/A140102(n) = t^2 = 3.38297576... (cf. A276800) where the limit of A140101(n)/A140100(n) = t = 1.839286755.. and t = tribonacci constant satisfies: t^3 = 1 + t + t^2.

Terms computed by Reinhard Zumkeller

A276801 Decimal expansion of t^3, where t is the tribonacci constant A058265.

Original entry on oeis.org

6, 2, 2, 2, 2, 6, 2, 5, 2, 3, 1, 2, 0, 3, 9, 8, 6, 2, 6, 6, 7, 4, 5, 6, 1, 1, 0, 1, 1, 0, 8, 3, 2, 1, 1, 8, 7, 3, 7, 3, 5, 6, 0, 7, 8, 9, 8, 4, 6, 1, 6, 8, 4, 2, 8, 7, 9, 8, 3, 2, 1, 3, 1, 6, 6, 3, 9, 5, 7, 5, 1, 1, 8, 0, 9, 1, 9, 0, 6, 7, 1, 7, 9, 6, 2, 0, 2, 8, 7, 5, 3, 4, 3, 2, 6, 7, 3, 1, 5, 3, 7, 4, 6, 0, 8, 0, 4

Offset: 1

N. J. A. Sloane, Oct 28 2016

A cubic integer with minimal polynomial x^3 - 7x^2 + 5x - 1, of which it is the unique real root. - Charles R Greathouse IV, Nov 06 2016

			6.222262523120398626674561101108321187373560789846168428798321316639575...

Index entries for algebraic numbers, degree 3

Cf. A058265, A276800.

RealDigits[x /. FindRoot[x^3 - 7*x^2 + 5*x - 1, {x, 6}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, May 27 2023 *)

polrootsreal(x^3-7*x^2+5*x-1)[1] \\ Charles R Greathouse IV, Nov 06 2016

1/t + 1/t^2 + 1/t^3 = 1/A058265 + 1/A276800 + 1/A276801 = 1.
From Dimitri Papadopoulos, Nov 07 2023: (Start)
t^3 = (A276800^2 + 1)/2.
t^3 + 1/t^3 = t + 1/t + 4.
t^3 = (1/4)*(t + 1)^2*(t - 1)^2*(t^2 + 1). (End)

A277722 a(n) = floor(n*tau^2) where tau is the tribonacci constant (A058265).

Original entry on oeis.org

0, 3, 6, 10, 13, 16, 20, 23, 27, 30, 33, 37, 40, 43, 47, 50, 54, 57, 60, 64, 67, 71, 74, 77, 81, 84, 87, 91, 94, 98, 101, 104, 108, 111, 115, 118, 121, 125, 128, 131, 135, 138, 142, 145, 148, 152, 155, 158, 162, 165, 169, 172, 175, 179, 182, 186, 189, 192, 196, 199, 202, 206, 209, 213, 216, 219

Offset: 0

N. J. A. Sloane, Oct 30 2016

nonn

JungHwan Min, Table of n, a(n) for n = 0..10000
A. J. Hildebrand, Junxian Li, Xiaomin Li and Yun Xie, Almost Beatty Partitions, arXiv:1809.08690 [math.NT], 2018.

Cf. A058265, A158919, A276800, A277723.

A277722 := proc(n)
    a276800 :=  3.3829757679062374941227085364550345869493820437485761820195626772353718960099402922235933340043661396041006 ;
    floor(n*a276800) ;
end proc:
seq(A277722(n),n=0..100) ; # R. J. Mathar, Nov 02 2016

A277722[n_] := Floor[n (1/3 (1 + (19 - 3 Sqrt[33])^(1/3) + (19 + 3 Sqrt[33])^(1/3)))^2]; Array[A277722, 66, 0] (* JungHwan Min, Nov 06 2016 *)

A276799 a(n) = floor(n*t^2) - A003145(n), where t = 1.8392867... is the tribonacci constant A058265.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Oct 28 2016

sign

a(n) is in {-1, 0, 1, 2}, but the first n for which -1 appears is n = 329. - Jeffrey Shallit, Nov 19 2016

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A003144, A003145, A003146, A275926, A058265, A276800, A277721, A278352 (positions of -1's).

a(n) = A277722(n) - A003145(n). - R. J. Mathar, Nov 02 2016

A286998 0-limiting word of the morphism 0->10, 1->20, 2->0.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 22 2017

Starting with 0, the first 5 iterations of the morphism yield words shown here:
  1st: 10
  2nd: 2010
  3rd: 0102010
  4th: 1020100102010
  5th: 201001020101020100102010
The 2-limiting word is the limit of the words for which the number of iterations is congruent to 2 mod 3.
Let U, V, W be the limits of u(n)/n, v(n)/n, w(n)/n, respectively. Then 1/U + 1/V + 1/W = 1, where
  U = 1.8392867552141611325518525646532866..., (A058265)
  V = U^2 = 3.3829757679062374941227085364..., (A276800)
  W = U^3 = 6.2222625231203986266745611011.... (A276801)
If n >=2, then u(n) - u(n-1) is in {1,2}, v(n) - v(n-1) is in {2,3,4}, and w(n) - w(n-1) is in {4,6,7}.
From Jiri Hladky, Aug 29 2021: (Start)
This is also Arnoux-Rauzy word sigma_0 x sigma_1 x sigma_2, where sigmas are defined as:
  sigma_0 : 0 -> 0,  1 -> 10, 2 -> 20;
  sigma_1 : 0 -> 01, 1 -> 1,  2 -> 21;
  sigma_2 : 0 -> 02, 1 -> 12, 2 -> 2.
Fixed point of the morphism 0->0102010, 1->102010, 2->2010, starting from a(1)=0. This definition has the benefit that EACH iteration yields the prefix of the limiting word.
Frequency of letters:
  0: 1/t   ~ 54.368% (A192918)
  1: 1/t^2 ~ 29.559%
  2: 1/t^3 ~ 16.071%
where t is tribonacci constant A058265.
Equals A347290 with a re-mapping of values 1->2, 2->1.
(End)

			3rd iterate: 0102010
6th iterate: 01020101020100102010201001020101020100102010

Jiri Hladky, Table of n, a(n) for n = 1..20000 (terms 1..10000 from Clark Kimberling).
L. Balková, M. Bucci, A. De Luca, J. Hladký, and S. Puzynina: Aperiodic Pseudorandom Number Generators Based on Infinite Words, Theoret. Comput. Sci. 647 (2016), 85-100.
Julien Cassaigne, Sebastien Ferenczi, and Luca Q. Zamboni, Imbalances in Arnoux-Rauzy sequences, Annales de l'institut Fourier, 50 (2000), 1265-1276.
D. Damanik and L. Q. Zamboni, Arnoux-Rauzy subshifts: linear recurrence, powers and palindromes, arXiv:math/0208137 [math.CO], 2002.
J. Patera, GENERATING THE FIBONACCI CHAIN IN O (log n) SPACE AND O (n) TIME (2003)
Gérard Rauzy, Nombres algébriques et substitutions, Bull. Soc. Math. France 110.2 (1982): 147-178.
Wikipedia, Rauzy fractal
Index entries for sequences that are fixed points of mappings

Cf. A080843, A286999, A287000, A287001, A287112, A287174, A347290 (values 0,2,1).

s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 0}, 2 -> 0}] &, {0}, 9] (* A286998 *)
Flatten[Position[s, 0]] (* A286999 *)
Flatten[Position[s, 1]] (* A287000 *)
Flatten[Position[s, 2]] (* A287001 *)
(* Using the 0->0102010, 1->102010, 2->2010 rule: *)
Nest[ Flatten[# /. {0 -> {0, 1, 0, 2, 0, 1, 0}, 1 -> {1, 0, 2, 0, 1, 0}, 2 -> {2, 0, 1, 0}}] &, {0}, 3]

A277724 Intersection of A158919 and A277722.

Original entry on oeis.org

0, 3, 16, 20, 23, 27, 33, 40, 47, 57, 60, 64, 71, 77, 84, 91, 101, 104, 108, 115, 121, 125, 128, 145, 148, 152, 158, 165, 169, 172, 182, 189, 196, 202, 206, 209, 213, 226, 233, 240, 246, 250, 253, 257, 263, 270, 274, 277, 290, 294, 297, 301, 307, 314, 321, 331, 334, 338, 345, 351, 358, 375, 378, 382

Offset: 1

N. J. A. Sloane, Oct 30 2016

nonn

See A277728 for discussion.

Cf. A003144, A003145, A003146, A058265, A158919, A275926, A276799, A276800, A277722, A277723, A277724, A277725, A277726, A277727, A277728.

A277725 Intersection of A158919 and A277723.

Original entry on oeis.org

0, 12, 18, 31, 49, 62, 68, 80, 93, 99, 112, 130, 136, 143, 161, 174, 180, 211, 217, 224, 242, 248, 255, 261, 286, 292, 323, 329, 336, 342, 354, 360, 367, 373, 404, 410, 423, 435, 441, 448, 454, 472, 485, 491, 516, 522, 535, 553, 560, 566, 572, 584, 597, 603, 616, 634, 640, 647, 665, 678, 684, 709, 715

Offset: 1

N. J. A. Sloane, Oct 30 2016

nonn

See A277728 for discussion.

Cf. A003144, A003145, A003146, A058265, A158919, A275926, A276799, A276800, A277722, A277723, A277724, A277725, A277726, A277727, A277728.

A277726 Intersection of A277722 and A277723.

Original entry on oeis.org

0, 6, 37, 43, 74, 87, 118, 155, 186, 192, 199, 230, 236, 267, 280, 304, 311, 317, 348, 385, 392, 416, 429, 460, 466, 497, 504, 510, 541, 578, 622, 659, 690, 696, 703, 734, 740, 771, 784, 808, 815, 852, 889, 896, 920, 933, 964, 970, 1001, 1008, 1014, 1045, 1082, 1126, 1163, 1194, 1200, 1207, 1238, 1244, 1275, 1288, 1312, 1319, 1356, 1387, 1393, 1400, 1424

Offset: 1

N. J. A. Sloane, Oct 30 2016

nonn

See A277728 for discussion.

Cf. A003144, A003145, A003146, A058265, A158919, A275926, A276799, A276800, A277722, A277723, A277724, A277725, A277726, A277727, A277728.

Digits := 120;
isA277722 := proc(n)
    a276800 :=  3.3829757679062374941227085364550345869493820437485761820195626772353718960099402922235933340043661396041006 ;
    for x from floor((n-3)/a276800) to (n+3)/a276800 do
        if floor(x*a276800) = n then
            return true;
        end if;
    end do:
    return false;
end proc:
isA277723 := proc(n)
    a276801 :=  6.2222625231203986266745611011083211873735607898461684287983213166395751180919067179620287534326731537460804;
    for x from floor((n-3)/a276801) to (n+3)/a276801 do
        if floor(x*a276801) = n then
            return true;
        end if;
    end do:
    return false;
end proc:
isA277726 := proc(n)
    isA277722(n) and isA277723(n) ;
end proc:
for n from 0 to 8000 do
    if isA277726(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Nov 02 2016

Corrected by R. J. Mathar, Nov 01 2016

cp's OEIS Frontend

A058265 Decimal expansion of the tribonacci constant t, the real root of x^3 - x^2 - x - 1.

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A003145 Positions of letter b in the tribonacci word abacabaabacababac... generated by a->ab, b->ac, c->a (cf. A092782).

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A140103 Term-by-term sums of A140101 and A140100; also, equals the complement of A140102, which is the term-by-term differences of A140101 and A140100, where A140101 is the complement of A140100.

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A276801 Decimal expansion of t^3, where t is the tribonacci constant A058265.

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A277722 a(n) = floor(n*tau^2) where tau is the tribonacci constant (A058265).

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A276799 a(n) = floor(n*t^2) - A003145(n), where t = 1.8392867... is the tribonacci constant A058265.

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A286998 0-limiting word of the morphism 0->10, 1->20, 2->0.

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