A105424 The part of n in base phi left of the decimal point, using a greedy algorithm representation (more precisely, using the Bergman-canonical representation).

0, 1, 10, 100, 101, 1000, 1010, 10000, 10001, 10010, 10100, 10101, 100000, 100010, 100100, 100101, 101000, 101010, 1000000, 1000001, 1000010, 1000100, 1000101, 1001000, 1001010, 1010000, 1010001, 1010010, 1010100, 1010101, 10000000

Offset: 0

Author

Bryan Jacobs (bryanjj(AT)gmail.com), Apr 08 2005

Keywords

• nonn
• base

Examples

			2 = 10.01 in base phi, so left of the decimal point is 10.
The first few numbers written in base phi:
0 = 0.
1 = 1.
2 = 10.01
3 = 100.01
4 = 101.01
5 = 1000.1001
6 = 1010.0001
7 = 10000.0001
8 = 10001.0001
9 = 10010.0101
10 = 10100.0101
11 = 10101.0101
12 = 100000.101001
13 = 100010.001001
14 = 100100.001001
15 = 100101.001001
16 = 101000.100001
17 = 101010.000001
18 = 1000000.000001
19 = 1000001.000001
20 = 1000010.010001
21 = 1000100.010001
22 = 1000101.010001
23 = 1001000.100101
24 = 1001010.000101
...
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• F. Michel Dekking, How to add two natural numbers in base phi, arXiv:2002.01665 [math.NT], 5 Feb 2020.
• Ron Knott, Phigits and the Base Phi representation.
• Ron Knott, Phigits and the Base Phi representation [Local copy, pdf only]
• Jeffrey Shallit, Proving Properties of phi-Representations with the Walnut Theorem-Prover, arXiv:2305.02672 [math.NT], 2023. [Note that this document has been revised multiple times.]

Crossrefs

Cf. A001622, A055778, A105425, A104605.

See A341722 for the part to the right of the decimal point.

Cf. A105116 (base e), A344939 (base Pi).

Programs

Mathematica

nn = 1000; len = 2*Ceiling[Log[GoldenRatio, nn]]; Table[d = RealDigits[n, GoldenRatio, len]; FromDigits[Take[d[[1]], d[[2]]]], {n, 0, nn}] (* T. D. Noe, May 20 2011 *)

Extensions

Definition clarified by N. J. A. Sloane, May 27 2023