cp's OEIS Frontend

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

%I A362692 #38 Jan 05 2025 19:51:42
%S A362692 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,
%T A362692 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,
%U A362692 9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10
%N A362692 Length of the "integer part" of the phi-expansion of n.
%C A362692 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.
%C A362692 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).
%H A362692 Paolo Xausa, <a href="/A362692/b362692.txt">Table of n, a(n) for n = 0..10000</a>
%H A362692 George Bergman, <a href="https://math.berkeley.edu/~gbergman/papers/base_tau.pdf">A number system with an irrational base</a>, Math. Mag. 31 (1957), 98-110.
%H A362692 G. R. Sanchis and L. A. Sanchis, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/39-2/sanchis.pdf">On the frequency of occurrence of α^i in the α-expansions of the positive integers</a>, Fibonacci Quart. 39 (2001), 123-137.
%H A362692 Jeffrey Shallit, <a href="https://arxiv.org/abs/2305.02672">Proving Properties of phi-Representations with the Walnut Theorem-Prover</a>, arXiv:2305.02672 [math.NT], 2023.
%F A362692 There is a linear representation of rank 9 for a(n).
%F A362692 a(n) = ceiling(log_phi(n)) for n >= 2.
%e A362692 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.
%t A362692 A362692[n_]:=Floor[Log[GoldenRatio,Max[n,1]]]+1;Array[A362692,100,0] (* _Paolo Xausa_, Oct 19 2023 *)
%Y A362692 Cf. A001622, A105424, A362716, A341722, A362872, A362917.
%K A362692 nonn
%O A362692 0,3
%A A362692 _Jeffrey Shallit_, May 01 2023
%E A362692 a(0) changed to 1 by _N. J. A. Sloane_, May 26 2023