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

0,3

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 "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).

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.

Formula

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

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

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A072073 A320757 A061716 * A341053 A126236 A198194

Adjacent sequences: A362689 A362690 A362691 * A362693 A362694 A362695

Keyword

nonn

Author

Jeffrey Shallit, May 01 2023

Extensions

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

Status

approved