OFFSET

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From Michel Dekking, Apr 19 2019: (Start):

Here is a proof of Velin Yanev's conjecture.

There is an offset problem. We will give the period-doubling sequence offset 1, and will also consider the characteristic function of powers of 2, A209229, instead of the Fredholm-Rueppel sequence A036987.

Let alpha be the period doubling morphism

alpha: 0 -> 01, 1 -> 00.

Let PD = 01000101... be the fixed point of alpha, with offset 1, so PD(n+1) = A096268(n).

Let beta be the time reversal of alpha

beta: 0 -> 10, 1 -> 00.

By definition, (a(n)) satisfies (a(n)) = 1 beta((a(n))), implying that

(*) (a(n)) = 1 beta(1) beta^2(1) beta^3(1) ....

Note that Yanev's conjecture can be reformulated as

/ PD(n) if n is not a power of 2,

a(n) = -|

\ 1-PD(n) if n is a power of 2.

It is easy to see that for all n

beta^n(0) = beta^n(1), EXCEPT for the first letter:

beta^n(0) = delta(n)R, beta^n(1) = (1-delta(n))R, where

delta(2n) = 0, delta(2n+1) = 1.

We will use below that, similarly, alpha^n(0) = alpha^n(1), EXCEPT for the LAST letter.

From (*) we see now that the conjecture will be proved if one shows that

(**) alpha^n(0) = 0 beta(0)beta^2(0)...beta^{n-1}(0) delta(n),

since the first letters of the beta^(k) in this concatenation occur exactly at positions 2^k.

The formula (**) has a companion

(***) beta^n(0) = delta(n) 0 beta(0)beta^2(0)...beta^{n-1}(0).

We will prove both formula's together with induction on n.

For n=2 (**) and (***) one has to check

alpha^2(0)=0beta(0)delta(2)=0100, beta^2(0)=delta(2)0beta(0)=0010,

which are true. Suppose (**), (***) hold for some n>1. Then

alpha^{n+1}(0) = alpha^n(0) alpha^n(1) =

0 beta(0)beta^2(0)...beta^{n-1}(0) delta(n)0

beta(0)beta^2(0)...beta^{n-1}(0) (1-delta(n)) =

0 beta(0)beta^2(0)...beta^n(0) (1-delta(n)) =

0 beta(0)beta^2(0)...beta^n(0)delta(n+1).

The formula for (***) with n+1 follows in a similar way.

(End)

MATHEMATICA

t = {1}; Do[If[EvenQ[n], a = 1 - t[[n/2]], a = 0]; AppendTo[t, a], {n, 2, 100}]; t (* T. D. Noe, Oct 10 2012 *)

PROG

CROSSREFS

KEYWORD

nonn

AUTHOR

John W. Layman, Oct 08 2012

EXTENSIONS

More terms from Antti Karttunen, Sep 30 2018

STATUS

approved