cp's OEIS Frontend

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