For every positive integer n, there is a unique finite sequence of distinct nonconsecutive (not necessarily positive) integers 
,
such that
 |
(1) |
where 
is the golden ratio.
For example, for the first few positive integers,
 |
 |
 |
(2) |
 |
|
 |
(3) |
 |
|
 |
(4) |
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|
 |
(5) |
 |
|
 |
(6) |
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|
 |
(7) |
 |
|
 |
(8) |
(Sloane's A104605).
The numbers of terms needed to represent n for n = 1, 2, ... are given by 1, 2, 2, 3, 3, 3, 2, 3, 4, 4, 5, 4, ... (Sloane's A055778), which are also the numbers of 1s in the base- representation of n.
The following tables summarized the values of n that require exactly k powers of in their representations.
| k | Sloane | |
| 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, ... |
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. Reading, MA: Addison-Wesley, 1997.
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." http://www.research.att.com/~njas/sequences/.