OFFSET
1,1
COMMENTS
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.
Note that a, b and c form a complementary triple (since consecutive digits 11 do not occur in a base phi representation).
Conjecture (Moses 2012/Baruchel 2018): b is the generalized Beatty sequence b(n) = floor(n*phi) + 2*n + 1.
One can prove that the Moses/Baruchel conjecture and the Kimberling conjecture are equivalent.
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.
This conjecture is compatible with the Moses/Baruchel/Kimberling conjecture.
These three conjectures are proved in my paper 'Base phi representations and golden mean beta-expansions'. - Michel Dekking, Jun 26 2019
LINKS
J.-P. Allouche, F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018.
George Bergman, A Number System with an Irrational Base, Mathematics Magazine, Vol. 31, No. 2 (Nov. - Dec., 1957), pp. 98-110.
M. Dekking, Base phi representations and golden mean beta-expansions, arXiv:1906.08437 [math.NT], 2019.
EXAMPLE
3 = phi^2 + phi^{-2}, 5 = phi^3 + phi^{-1} + phi^{-4}.
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Dekking, May 06 2019
STATUS
approved