A362716 -id:A362716 - cp's OEIS frontend

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

Jeffrey Shallit, May 01 2023

nonn

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).

			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.

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.

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

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

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

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

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

A362756 Sum of the bits of the "fractional part" of the base-phi representation of n.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 4, 3, 3, 3, 3, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 3, 2, 2, 2, 3, 3, 3, 4, 3, 3, 3, 3, 2, 2, 2, 3, 3, 3, 4, 3, 3, 3, 4, 4, 4, 5, 4, 4, 4, 4, 3, 3, 3, 4, 4

Offset: 0

Jeffrey Shallit, May 02 2023

nonn

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(-1).

The first difference of a(n) is Fibonacci-automatic and takes values in {-1,0,1} only.

			For n = 20 the phi-representation is 1000010.010001, so a(20) = 2.

George Bergman, A number system with an irrational base, Math. Mag. 31 (1957), 98-110.

Cf. A055778, A362716.

There is a linear representation of rank 21 for a(n).

cp's OEIS Frontend

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

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