Phi Number System
For every positive integer n, there is a unique finite sequence of distinct nonconsecutive (not necessarily positive) integers k_1, ..., k_m such that
| n=phi^(k_1)+...+phi^(k_m), |
(1)
|
where phi is the golden ratio.
For example, for the first few positive integers,
1 = phi^0
(2)
2 = phi^(-2)+phi
(3)
3 = phi^(-2)+phi^2
(4)
4 = phi^(-2)+phi^0+phi^2
(5)
5 = phi^(-4)+phi^(-1)+phi^3
(6)
6 = phi^(-4)+phi+phi^3
(7)
7 = phi^(-4)+phi^4
(8)
(OEIS 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, ... (OEIS A055778), which are also the numbers of 1s in the base-phi representation of n.
The following tables summarizes the values of n that require exactly k powers of phi in their representations.
See also
Base, Golden RatioExplore with Wolfram|Alpha
WolframAlpha
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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. 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."Referenced on Wolfram|Alpha
Phi Number SystemCite this as:
Weisstein, Eric W. "Phi Number System." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PhiNumberSystem.html