A341722 The part of n in base phi right of the decimal point (reversed), using a greedy algorithm representation (more precisely, using the Bergman-canonical representation).

0, 0, 10, 10, 10, 1001, 1000, 1000, 1000, 1010, 1010, 1010, 100101, 100100, 100100, 100100, 100001, 100000, 100000, 100000, 100010, 100010, 100010, 101001, 101000, 101000, 101000, 101010, 101010, 101010, 10010101, 10010100, 10010100, 10010100, 10010001, 10010000, 10010000

Offset

0,3

Comments

A105424 and A105425 give the part of n in base phi left of the decimal point.

Links

Hugo Pfoertner, 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.

C. Frougny and J. Sakarovitch, Automatic conversion from Fibonacci representation to representation in base phi, and a generalization, Int. J. Algebra Comput. 9 (1999), 351-384. See also preprint.

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.

Example

The first few numbers written in base phi are:
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
...

Crossrefs

Cf. A105424, A105425.

Sequence in context: A003855 A245431 A377216 * A229902 A004209 A162720

Adjacent sequences: A341719 A341720 A341721 * A341723 A341724 A341725

Keyword

nonn,base,easy

Author

N. J. A. Sloane, Mar 01 2021

Extensions

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

Status

approved