A306683 - OEIS

%I #39 Jun 26 2019 01:01:25

%S 3,5,7,10,12,14,16,18,21,23,25,28,30,32,34,36,39,41,43,45,47,50,52,54,

%T 57,59,61,63,65,68,70,72,75,77,79,81,83,86,88,90,92,94,97,99,101,104,

%U 106,108,110,112,115,117,119,121,123,126,128,130,133,135,137,139,141,144

%N Integers k for which the base-phi representation of k does not include 1 or phi.

%C Let b = A214970 be the sequence of the integers k for which the base phi representation includes 1, and let c be the sequence of integers k for which the base phi representation includes phi.

%C Note that a, b and c form a complementary triple (since consecutive digits 11 do not occur in a base phi representation).

%C Conjecture (Moses 2012/Baruchel 2018): b is the generalized Beatty sequence b(n) = floor(n*phi) + 2*n + 1.

%C Conjecture (Kimberling 2012): c = A054770 = A000201(n) + 2*n - 1 = floor(n*phi) + 2*n - 1.

%C One can prove that the Moses/Baruchel conjecture and the Kimberling conjecture are equivalent.

%C Conjecture: (a(n)) is a union of two generalized Beatty sequences v and w, given by v(n) = floor(n*phi) + 2*n = A003231(n), and w(n) = 3*floor(n*phi) + n + 1 = A190509(n) + 1.

%C This conjecture is compatible with the Moses/Baruchel/Kimberling conjecture.

%C These three conjectures are proved in my paper 'Base phi representations and golden mean beta-expansions'. - _Michel Dekking_, Jun 26 2019

%H J.-P. Allouche, F. M. Dekking, <a href="https://arxiv.org/abs/1809.03424">Generalized Beatty sequences and complementary triples</a>, arXiv:1809.03424 [math.NT], 2018.

%H George Bergman, <a href="https://www.jstor.org/stable/3029218">A Number System with an Irrational Base</a>, Mathematics Magazine, Vol. 31, No. 2 (Nov. - Dec., 1957), pp. 98-110.

%H M. Dekking, <a href="https://arxiv.org/abs/1906.08437">Base phi representations and golden mean beta-expansions</a>, arXiv:1906.08437 [math.NT], 2019.

%e 3 = phi^2 + phi^{-2}, 5 = phi^3 + phi^{-1} + phi^{-4}.

%Y Cf. A214970, A054770, A000201.

%K nonn

%O 1,1

%A _Michel Dekking_, May 06 2019