A284796 Positions of 1's in A284793.

1, 4, 7, 9, 12, 16, 19, 22, 25, 28, 31, 33, 36, 40, 43, 45, 48, 52, 55, 58, 61, 63, 66, 70, 73, 76, 79, 81, 84, 88, 91, 94, 97, 100, 103, 105, 108, 112, 115, 117, 120, 124, 127, 130, 133, 136, 139, 141, 144, 148, 151, 153, 156, 160, 163, 166, 169, 171, 174

Offset: 1

Author

Clark Kimberling, Apr 14 2017

Keywords

• nonn
• easy

Comments

This sequence and A284794 and A284795 form a partition of the positive integers. Conjecture: for n>=1, we have a(n)-3n+3 in {0,1}, 3n+2-A284795(n) in {0,1,2,3}, and 3n-2-A284795(n) in {0,1}.

A284793 = (1,-1,0,1,0,-1,1,-1,1,-1,0,1,0,-1,0,1,0,-1,1,-1,0,1,0,-1, ... ); thus

A284794 = (2,6,8,10,14,...)

A284795 = (3,5,11,13,15,...)

A284796 = (1,4,7,9,12,15,...).

From Michel Dekking, Nov 24 2019: (Start)

Here is a proof of Kimberling's conjecture, i.e., the sequence y defined by y(n) := a(n)-3n+3 takes only values in the alphabet {0,1}. We know that A284793 = 1,-1,0,1,0,-1,... is a morphic sequence(see A284793). Let tau on the alphabet {A,B,C,D} be given by

A -> BC, B->BC, C->ABDC, D->ABDC.

The unique fixed point of tau is x = BCABDCBC... The letter-to letter map pi which gives A284793 = pi(x) is given by

pi(A)=0, pi(B)=1, pi(C)=-1, pi(D)=0.

The return words of B (i.e., the words with prefix B and no other occurrences of B) in x are

a:= BCA, b:= BDC, c:= BC, d:= BDCA.

The morphism tau induces a so-called derivated morphism on the alphabet of return words, which is given by

beta(a) = abc, beta(b) = adb, beta(c) = ab, beta(d) = adbc.

Since B is the unique letter in {A,B,C,D} projecting on the letter 1, the difference sequence Delta*(a(n)) is given by replacing a,b,c,d by their lengths in the fixed point abcadbab... of beta:

a->3, b->3, c->2, d->4.

The difference sequence (Delta (y(n)) is given by

y(n+1)-y(n) = a(n+1)-a(n)-3.

It follows that Delta y only takes the values 0, -1 and 1. Moreover, the 4 words a,b,c,d have projections

pi(BCA)=1,-1,0; pi(BDC)=1,0,-1; pi(BC)=1,-1; pi(BDCA)=1,0,-1,0.

From this we see that 1 and -1 always occur in pairs with 1 first, within the 4 projections of a,b,c, and d. Since y(1)=1, this implies that y itself takes only values in {0,1}.

(End)

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A284793, A284794, A284795, A189664.

Programs

Mathematica

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 0, 1, 1}}] &, {0}, 7] (* A284775 *)
d = Differences[s]  (* A284793 *)
Flatten[Position[d, -1]] (* A284794 *)
Flatten[Position[d, 0]]  (* A284795 *)
Flatten[Position[d, 1]]  (* A284796 *)
d1/2  (* positions of 0 in A189664 *)