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

0,3

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).

Links

Table of n, a(n) for n=0..83.

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.

Formula

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

Example

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

Crossrefs

Cf. A341722, A362692.

Sequence in context: A219654 A096491 A217871 * A306390 A106160 A305117

Adjacent sequences: A362869 A362870 A362871 * A362873 A362874 A362875

Keyword

nonn

Author

Jeffrey Shallit, May 07 2023

Status

approved