cp's OEIS Frontend

"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

Comment from Philippe Deléham, Feb 27 2009: 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.

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 c 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 11. [Duchene and Rigo, Remark 2.5] - N. J. A. Sloane, Mar 02 2019

The minimal polynomial of this constant is x^3 - 3*x^2 - x - 1, and it is its unique real root. - Amiram Eldar, May 27 2023

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)

See A277728 for discussion.

From Clark Kimberling, May 21 2017: (Start)

Let u be the sequence of positions of 0, and likewise, v for 1 and w for 2. 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...,

V = U^2 = 3.3829757679062374941227085364...,

W = U^3 = 6.2222625231203986266745611011....

If n >=2, then u(n) - u(n-1) is in {1,2,3}, v(n) - v(n-1) is in {2,4,5}, and w(n) - w(n-1) is in {4,7,9}. (u = A277736, v = A277737, w = A277738). (End)

Although I believe the assertions in Kimberling's comment above to be correct, these results are quite tricky to prove, and unless a formal proof is supplied at present these assertions must be regarded as conjectures. - N. J. A. Sloane, Aug 20 2018

From Michel Dekking, Oct 03 2019: (Start)

Here is a proof of Clark Kimberling's conjectures (and more).

The incidence matrix of the defining morphism

sigma: 0 -> 01, 1 -> 20, 2 -> 0

is the same as the incidence matrix of the tribonacci morphism

0 -> 01, 1 -> 02, 2 -> 0

(see A080843 and/or A092782).

This implies that the frequencies f0, f1 and f2 of the letters 0,1, and 2 in (a(n)) are the same as the corresponding frequencies in the tribonacci word, which are 1/t, 1/t^2 and 1/t^3 (see, e.g., A092782).

Since U = 1/f0, V = 1/f1, and W = 1/f2, we conclude that

U = t = A058265, V = t^2 = A276800 and W = t^3 = A276801.

The statements on the difference sequences u, v, and w of the positions of 0,1, and 2 are easily verified by applying sigma to the return words of these three letters.

Here the return words of an arbitrary word w in a sequence x are all the words occurring in x with prefix w that do not have other occurrences of w in them.

The return words of 0 are 0, 01, and 012, which indeed have length 1, 2

and 3. Since

sigma(0) = 01, sigma(1) = 0120, and sigma(012) = 01200,

one sees that u is the unique fixed point of the morphism

1 -> 2, 2-> 31, 3 ->311.

With a little more work, passing to sigma^2, and rotating, one can show that v is the unique fixed point of the morphism

2->52, 4->5224, 5->52244 .

Similarly, w is the unique fixed point of the morphism

4->94, 7->9447, 9->94477.

Interestingly, the three morphisms having u,v, and w as fixed point are essentially the same morphism (were we replaced sigma by sigma^2) with standard form

1->12, 2->1223, 3->12233.

(End)

The kind of phenomenon observed at the end of the previous comment holds in a very strong way for the tribonacci word. See Theorem 5.1. in the paper by Huang and Wen. - Michel Dekking, Oct 04 2019

Starting with 0, the first 4 iterations of the morphism yield words shown here:

1st: 10

2nd: 2010

3rd: 0102010

4th: 1020100102010

The 1-limiting word is the limit of the words for which the number of iterations is congruent to 1 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...,

V = U^2 = 3.3829757679062374941227085364...,

W = U^3 = 6.2222625231203986266745611011....

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 Michel Dekking, Mar 29 2019: (Start)

This sequence is one of the three fixed points of the morphism alpha^3, where alpha is the defining morphism

0->10, 1->20, 2->0.

The other two fixed points are A286998 and A287174.

We have alpha = rho(tau), where tau is the tribonacci morphism in A080843

0->01, 1->02, 2->0,

and rho is the rotation operator.

An eigenvector computation of the incidence matrix of the morphism gives that 0,1, and 2 have frequencies 1/t, 1/t^2 and 1/t^3, where t is the tribonacci constant A058265.

Apparently (u(n)) := A287113. Thus U, the limit of u(n)/n, equals 1/(1/t), the tribonacci constant t. Also, V = A276800, and W = A276801.

(End)

Always in the set {-1,0,1,2,3}, but first occurrence of -1 is at n = 2047. - Jeffrey Shallit, Nov 19 2016

Because the tribonacci constant t = A058265 > 1, with Beatty sequence At(n) := floor(n*t), n >= 1 (with At(0) = 0) given in A158919, has the companion sequence Bt := floor(n*s), n >= 1, (with Bt(0) = 0), with 1/t + 1/s = 1, and At and Bt are complementary, disjoint sequences for the positive integers. Note that Bt is not A172278. The first entries n = 0..161 coincide. A172278(162) = 354 but At(193) = A158919(193) = 354, hence A172278 is not complementary together with At. In fact, Bt(162) = 355, which is not a member of At.

s-1 = 1/(t-1) equals the real root of 2*x^3 - 2*x - 1. See the formulas below. - Wolfdieter Lang, Sep 15 2022