|
%I #37 Feb 05 2023 01:59:57
%S 6,9,10,14,15,22,23,35,36,46,53,56,57,67,74,75,82,85,86,90,91,101,108,
%T 109,116,119,120,129,132,133,137,138,145,146,156,163,164,171,174,175,
%U 184,187,188,192,193,205,208,209,213,214,221,222,234,235,245,252,253,260,263,264,273
%N Numbers k such that both k and k+1 are in A095096.
%C From _Michel Dekking_, Oct 09 2020: (Start)
%C 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.
%C 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.
%C To this end one considers the 2-block substitution sigma^[2] of the Zeckendorf morphism
%C sigma: 1->12, 2->4, 3->1, 4->43.
%C 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.
%C Coding the words of length 2 in lexicographic order this gives sigma^[2] on the alphabet {1,2,...,7,8} as
%C sigma^[2]: 1->23, 2->24, 3->7, 4->8, 5->1, 6->2, 7->75, 8->76.
%C 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).
%C 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].
%C An eigenvalue calculation then gives the value sqrt(5)/10 from above.
%C Final remark: the same result has been derived for the base-phi expansion of the natural numbers, and the limit is the same.
%C (End)
%D Anton Shutov, On the sum of digits of the Zeckendorf representations of two consecutive numbers, Fib. Q., 58:3 (2020), 203-207.
%H Amiram Eldar, <a href="/A337287/b337287.txt">Table of n, a(n) for n = 1..10000</a>
%H F. Michel Dekking, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Dekking/dekk4.html">Morphisms, Symbolic Sequences, and Their Standard Forms</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
%H Michel Dekking, <a href="https://arxiv.org/abs/1911.10705">The sum of digits function of the base phi expansion of the natural numbers</a>,arXiv:1911.10705 [math.NT], 2019.
%t SequencePosition[Mod[DigitCount[Select[Range[0, 3000], BitAnd[#, 2 #] == 0 &], 2, 1], 2], {0, 0}][[;; , 1]] - 1 (* _Amiram Eldar_, Feb 05 2023 *)
%Y Cf. A020899, A095076, A095096, A337288, A337289, A337290, A337634, A337635, A337636, A337637.
%K nonn,base
%O 1,1
%A _N. J. A. Sloane_, Sep 12 2020
|
