For every positive integer integers
(1)
where golden ratio .
For example, for the first few positive integers,
(OEIS A104605 ).
The numbers of terms needed to represent A055778 ), which are also the numbers
of 1s in the base-
The following tables summarizes the values of
OEIS numbers requiring
exactly 2 A005248 2, 3, 7, 18, 47, 123, 322, 843, ... 3 A104626 4,
5, 6, 8, 19, 48, 124, 323, 844, ... 4 A104627 9,
10, 12, 13, 14, 16, 17, 20, 21, 25, ... 5 A104628 11,
15, 22, 23, 24, 26, 30, 31, 32, 34, ...
See also Base ,
Golden
Ratio
Explore with Wolfram|Alpha
References Bergman, G. "A Number System with an Irrational Base." Math. Mag. 31 , 98-110, 1957. Knott, R. "Using Powers
of Phi to represent Integers (Base Phi)." http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phigits.html . Knuth,
D. The
Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. Levasseur, K. "The Phi Number
System." http://www.hostsrv.com/webmaa/app1/MSP/webm1010/PhiNumberSystem/PhiNumberSystem.msp . Rousseau,
C. "The Phi Number System Revisited." Math. Mag. 68 , 283-284,
1995. Sloane, N. J. A. Sequences A005248 /M0848,
A055778 , A104605 ,
A104626 , A104627 ,
and A104628 in "The On-Line Encyclopedia
of Integer Sequences." Referenced on Wolfram|Alpha Phi Number System
Cite this as:
Weisstein, Eric W. "Phi Number System."
From MathWorld https://mathworld.wolfram.com/PhiNumberSystem.html
Subject classifications
, there is a unique finite sequence of distinct nonconsecutive
(not necessarily positive) integers 
, ..., 
such that
is the golden ratio.
for 
, 2, ... are given by 1, 2, 2, 3, 3, 3, 2, 3, 4, 4, 5, 4,
... (OEIS A055778), which are also the numbers
of 1s in the base-
representation of 
.
that require exactly 
powers of 
in their representations.
powers