A362692 Length of the "integer part" of the phi-expansion of n.

1, 1, 2, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10

Offset: 0

Author

Jeffrey Shallit, May 01 2023

Keywords

• nonn

Comments

The phi-representation of n is the (essentially) unique way to write n = Sum_{j=L..R} b(j)*phi^j, where b(j) is in {0,1} and -oo < L <= 0 <= R, where phi = (1+sqrt(5))/2, subject to the condition that b(j)b(j+1) != 1. The "integer" part is the string of bits b(R)b(R-1)...b(1)b(0), and its length is thus R+1.

The gaps between consecutive terms are all either 0 or 1, and a gap of 1 occurs if and only if n = 1 or n = L(2i) or n = L(2i-1) + 1 for i >= 1. This is equivalent to Theorem 2.1 of Sanchis and Sanchis (2001).

Examples

			For n = 20 we have n = phi^6 + phi^1 + phi^(-2) + phi^(-6), and the "integer part" has largest term phi^6, so a(20) = 7.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000
• George Bergman, A number system with an irrational base, Math. Mag. 31 (1957), 98-110.
• G. R. Sanchis and L. A. Sanchis, On the frequency of occurrence of α^i in the α-expansions of the positive integers, Fibonacci Quart. 39 (2001), 123-137.
• Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023.

Crossrefs

Cf. A001622, A105424, A362716, A341722, A362872, A362917.

Programs

Mathematica

A362692[n_]:=Floor[Log[GoldenRatio,Max[n,1]]]+1;Array[A362692,100,0] (* Paolo Xausa, Oct 19 2023 *)

Formula

There is a linear representation of rank 9 for a(n).

a(n) = ceiling(log_phi(n)) for n >= 2.

Extensions

a(0) changed to 1 by N. J. A. Sloane, May 26 2023