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# Phi Number System

For every positive integer , 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) (5) (6) (7) (8)

(OEIS A104605).

The numbers of terms needed to represent 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 .

The following tables summarizes the values of that require exactly powers of in their representations.

 OEIS numbers requiring exactly powers 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, ...

Base, Golden Ratio

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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 System

## Cite this as:

Weisstein, Eric W. "Phi Number System." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PhiNumberSystem.html