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A319430 First differences of the tribonacci representation numbers (A003726 or A278038).

%I A319430 #37 Dec 23 2019 18:57:56
%S A319430 1,1,1,1,1,1,2,1,1,1,1,1,3,1,1,1,1,1,1,2,1,1,1,5,1,1,1,1,1,1,2,1,1,1,
%T A319430 1,1,3,1,1,1,1,1,1,10,1,1,1,1,1,1,2,1,1,1,1,1,3,1,1,1,1,1,1,2,1,1,1,5,
%U A319430 1,1,1,1,1,1,2,1,1,1,1,1,19,1,1,1,1,1,1,2,1,1,1,1,1,3,1,1,1,1,1,1,2,1,1,1,5
%N A319430 First differences of the tribonacci representation numbers (A003726 or A278038).
%C A319430 This sequence appears to consist of runs of 1's of lengths given (essentially) by A275925, separated by single numbers > 1, which define the terms of A319431.
%C A319430 It would be nice to have a recurrence of some kind that produces A319431.
%H A319430 Rémy Sigrist, <a href="/A319430/b319430.txt">Table of n, a(n) for n = 0..50000</a> (first 9999 terms from N. J. A. Sloane)
%F A319430 Conjecture: All terms are of the form ceiling(2^k/7) for some k (cf. A046630), and all numbers of the form ceiling(2^k/7) occur.
%F A319430 Conjecture (continued): Furthermore, new values of ceiling(2^k/7) (that is, new records) appear at n = 0, 6,12, 23, 43, 80, 148, 273, ..., which (apart from the start) are the tribonacci numbers minus 1, A000073 - 1, or A089068.
%F A319430 a(n) = ceiling(2^i/7) iff the Tribonacci representation of n+1 ends in i 0's. - _Jeffrey Shallit_, Oct 02 2018
%t A319430 Differences@ Select[Range[0, 160], SequenceCount[IntegerDigits[#, 2], {1, 1, 1}] == 0 &] (* _Michael De Vlieger_, Dec 23 2019 *)
%Y A319430 Cf. A000073, A003726, A046630, A089068, A275925, A278038, A319431.
%K A319430 nonn
%O A319430 0,7
%A A319430 _N. J. A. Sloane_, Sep 30 2018