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 1-3 of 3 results.

A277735 Unique fixed point of the morphism 0 -> 01, 1 -> 20, 2 -> 0.

Original entry on oeis.org

0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1
Offset: 1

Views

Author

N. J. A. Sloane, Nov 07 2016

Keywords

Comments

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

Links

Crossrefs

Cf. A277736, A277737, A277738.
Equals A100619(n)-1.

Programs

  • Maple
    with(ListTools);
T:=proc(S) Flatten(subs( {0=[0,1], 1=[2,0], 2=[0]}, S)); end;
S:=[0];
for n from 1 to 10 do S:=T(S); od:
S;
  • Mathematica
    s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {2, 0}, 2 -> 0}] &, {0}, 10] (* A277735 *)
Flatten[Position[s, 0]] (* A277736 *)
Flatten[Position[s, 1]] (* A277737 *)
Flatten[Position[s, 2]] (* A277738 *)
(* Clark Kimberling, May 21 2017 *)

Extensions

Name clarified by Michel Dekking, Oct 03 2019

A277736 Positions of 0's in A277735.

Original entry on oeis.org

1, 4, 5, 6, 8, 10, 13, 14, 17, 18, 21, 22, 23, 25, 28, 29, 30, 32, 35, 36, 37, 39, 41, 44, 45, 48, 49, 50, 52, 54, 57, 58, 61, 62, 63, 65, 67, 70, 71, 74, 75, 78, 79, 80, 82, 85, 86, 87, 89, 91, 94, 95, 98, 99, 102, 103, 104, 106, 109, 110, 111, 113, 115, 118, 119, 122, 123, 126, 127, 128, 130, 133, 134
Offset: 1

Views

Author

N. J. A. Sloane, Nov 07 2016

Keywords

Comments

{A277736, A277737, A277738} forms a three-way partition of the positive integers, similar to {A003144, A003145, A003146}.

Links

Crossrefs

Cf. A277735, A277737, A277738.
Cf. also A003144, A003145, A003146.

Programs

  • Maple
    with(ListTools);
T:=proc(S) Flatten(subs( {0=[0,1], 1=[2,0], 2=[0]}, S)); end;
S:=[0];
for n from 1 to 14 do S:=T(S); od:
S; # A277735
p0:=[]: p1:=[]: p2:=[]:
for i from 1 to nops(S) do
j:=S[i];
if j=0 then p0:=[op(p0),i];
elif j=1 then p1:=[op(p1),i];
else p2:=[op(p2),i]; fi: od:
p0; # A277736
p1; # A277737
p2: # A277738

A277737 Positions of 1's in A277735.

Original entry on oeis.org

2, 7, 9, 11, 15, 19, 24, 26, 31, 33, 38, 40, 42, 46, 51, 53, 55, 59, 64, 66, 68, 72, 76, 81, 83, 88, 90, 92, 96, 100, 105, 107, 112, 114, 116, 120, 124, 129, 131, 136, 138, 143, 145, 147, 151, 156, 158, 160, 164, 168, 173, 175, 180, 182, 187, 189, 191, 195, 200, 202, 204, 208, 212, 217, 219, 224, 226
Offset: 1

Views

Author

N. J. A. Sloane, Nov 07 2016

Keywords

Comments

{A277736, A277737, A277738} forms a three-way partition of the positive integers, similar to {A003144, A003145, A003146}.

Links

Crossrefs

Cf. A277735, A277736, A277738.
Cf. also A003144, A003145, A003146.

Programs

  • Maple
    with(ListTools);
T:=proc(S) Flatten(subs( {0=[0,1], 1=[2,0], 2=[0]}, S)); end;
S:=[0];
for n from 1 to 14 do S:=T(S); od:
S; # A277735
p0:=[]: p1:=[]: p2:=[]:
for i from 1 to nops(S) do
j:=S[i];
if j=0 then p0:=[op(p0),i];
elif j=1 then p1:=[op(p1),i];
else p2:=[op(p2),i]; fi: od:
p0; # A277736
p1; # A277737
p2: # A277738
Showing 1-3 of 3 results.

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