cp's OEIS Frontend

From Michel Dekking, Oct 09 2020: (Start)

Let s_Z = A095076 be the parity of the sum of digits function of the Zeckendorf representation. Shutov's main result is that the number of times that s_Z(k) mod 2 = 0 AND s_Z(k+1) mod 2 = 0 in [0,n] divided by n tends to sqrt(5)/10.

It is possible to derive this result in a few lines by using the representation of s_Z as a morphic sequence, as given in the Comments of A095076.

To this end one considers the 2-block substitution sigma^[2] of the Zeckendorf morphism

sigma: 1->12, 2->4, 3->1, 4->43.

There are 10 words of length 2 occurring in the fixed points of this morphism. These are 11, 12, 14, 21, 24, 31, 34, 41, 43 and 44. Since the sigma^[2]-images of both 12 and 14 are 12,24, and this is also the case for the pair 41 and 43, one can reduce the number of letters to 8.

Coding the words of length 2 in lexicographic order this gives sigma^[2] on the alphabet {1,2,...,7,8} as

sigma^[2]: 1->23, 2->24, 3->7, 4->8, 5->1, 6->2, 7->75, 8->76.

The letter-to-letter map lambda mapping the fixed point of sigma^[2] to the sequence s_Z = A095076 is given by lambda(1)=0, lambda(2)=1, lambda(3)=0, lambda(4)=1 (see A095076).

We see that lambda(11) = lambda(31) = 00, and these are the only words of length 2 mapping to 00. It follows that the frequency of 00 in s_Z is equal to the sum of the frequencies of 1 and 5 in the fixed point starting with 2 of the morphism sigma^[2]. It is well known that these frequencies are given by the normalized eigenvector corresponding to the Perron-Frobenius eigenvalue of the incidence matrix of the morphism sigma^[2].

An eigenvalue calculation then gives the value sqrt(5)/10 from above.

Final remark: the same result has been derived for the base-phi expansion of the natural numbers, and the limit is the same.

(End)