A362872 Length of the "fractional part" of the phi-representation of n.

0, 0, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10, 10

Offset: 0

Author

Jeffrey Shallit, May 07 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 "fractional" part is the string of bits b(L)b(L+1)...b(-1), and its length is thus L.

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

Examples

			The phi-representation of 20 is 1000010.010001, so a(20) = 6.
		

Links

• 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.

Crossrefs

Cf. A341722, A362692.

Formula

There is a linear representation of rank 11 for a(n).