A306683 Integers k for which the base-phi representation of k does not include 1 or phi.

3, 5, 7, 10, 12, 14, 16, 18, 21, 23, 25, 28, 30, 32, 34, 36, 39, 41, 43, 45, 47, 50, 52, 54, 57, 59, 61, 63, 65, 68, 70, 72, 75, 77, 79, 81, 83, 86, 88, 90, 92, 94, 97, 99, 101, 104, 106, 108, 110, 112, 115, 117, 119, 121, 123, 126, 128, 130, 133, 135, 137, 139, 141, 144

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.

Conjecture (Kimberling 2012): c = A054770 = A000201(n) + 2*n - 1 = 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

Table of n, a(n) for n=1..64.

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

Cf. A214970, A054770, A000201.

Sequence in context: A046868 A225240 A104309 * A184586 A190511 A260466

Adjacent sequences: A306680 A306681 A306682 * A306684 A306685 A306686

Keyword

nonn

Author

Michel Dekking, May 06 2019

Status

approved