A194584 Differences of A035336.

5, 3, 5, 5, 3, 5, 3, 5, 5, 3, 5, 5, 3, 5, 3, 5, 5, 3, 5, 3, 5, 5, 3, 5, 5, 3, 5, 3, 5, 5, 3, 5, 5, 3, 5, 3, 5, 5, 3, 5, 3, 5, 5, 3, 5, 5, 3, 5, 3, 5, 5, 3, 5, 3, 5, 5, 3, 5, 5, 3, 5, 3, 5, 5, 3, 5, 5, 3, 5, 3, 5, 5, 3, 5, 3, 5, 5, 3, 5, 5, 3, 5, 3, 5, 5, 3

Offset: 1

Author

John W. Layman, Aug 29 2011

Keywords

• nonn

Comments

It appears that this is the Fibonacci word A003849, using 5's and 3's instead of 0's and 1's. In other words, {a(n)} is a fixed point of the morphism 5->53, 3->5.

Proof of this conjecture: since A035336(n) = (2*floor(n*phi) + n - 1) (with phi = (1+sqrt(5))/2) is a generalized Beatty sequence, this follows from Lemma 4 in Allouche and Dekking. - Michel Dekking, Oct 10 2018

Also differences of A089910. - Bob Selcoe, Sep 20 2014

Proof of this conjecture: this follows from the Carlitz-Scoville-Hoggatt theorem: compositions of the Wythoff A and B sequences are generalized Beatty sequences (cf. Theorem 1 in Allouche and Dekking). - Michel Dekking, Oct 10 2018

Links

• J.-P. Allouche and F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018.

Crossrefs

Cf. A003849, A035336, A089910.

Programs

Mathematica

Table[2 Floor[n (1 + Sqrt[5])/2] + n - 1, {n, 1, 100}] // Differences (* Jean-François Alcover, Dec 14 2018 *)