%I #48 Aug 26 2022 02:49:14

%S 2,3,22,38,342,598,5462,9558,87382,152918,1398102,2446678,22369622,

%T 39146838

%N Records in A268865.

%C Conjecture 1: Let m(n) be the position in A268865 corresponding to a(n). Then m(n) = (2/3)*(4^n-1), if n is even, and m(n) = (2/3)*(4^(n-1)-1) + 3*4^(n-1), if n is odd.

%C Conjecture 2: a(n) = 2*m(n) + 2, if n is even, and a(n) = (7*m+12)/11, if n is odd.

%C If Conjectures 1 and 2 are true, then we easily have for even n >= 0, m(n) == 0 (mod 10), a(n) == 2 (mod 10), while a(1) = m(1) = 3 and for odd n >= 3, m(n), a(n) == 8 (mod 10). The sequence {m(n)} begins: 0, 3, 10, 58, 170, 938, 2730, 15018, 43690, ...

%H Jeffrey Shallit, Sonja Linghui Shan, and Kai Hsiang Yang, <a href="https://arxiv.org/abs/2208.06025">Automatic Sequences in Negative Bases and Proofs of Some Conjectures of Shevelev</a>, arXiv:2208.06025 [cs.FL], 2022.

%H Vladimir Shevelev, <a href="http://arxiv.org/abs/1603.04434">Two analogs of Thue-Morse sequence</a>, arXiv:1603.04434 [math.NT], 2016.

%t nn = 10^5; f[n_, b_] := Most@ Mod[NestWhileList[-(#1 - Mod[#1, b])/b &, n, #1 != 0 &], b]; Union@ FoldList[Max, Array[(k = 0; While[Mod[Total@ f[k, 2], 2] == ThueMorse[# + k], k++]; k) &, nn - 1, 0]] (* _Michael De Vlieger_, Aug 25 2022 *)

%Y Cf. A268865.

%K nonn,more

%O 0,1

%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Feb 15 2016