%I #35 May 27 2023 12:03:01
%S 0,0,10,10,10,1001,1000,1000,1000,1010,1010,1010,100101,100100,100100,
%T 100100,100001,100000,100000,100000,100010,100010,100010,101001,
%U 101000,101000,101000,101010,101010,101010,10010101,10010100,10010100,10010100,10010001,10010000,10010000
%N 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).
%C A105424 and A105425 give the part of n in base phi left of the decimal point.
%H Hugo Pfoertner, <a href="/A341722/b341722.txt">Table of n, a(n) for n = 0..1000</a>
%H F. Michel Dekking, <a href="https://arxiv.org/abs/2002.01665">How to add two natural numbers in base phi</a>, arXiv:2002.01665 [math.NT], 5 Feb 2020.
%H C. Frougny and J. Sakarovitch, <a href="https://doi.org/10.1142/S0218196799000230">Automatic conversion from Fibonacci representation to representation in base phi, and a generalization</a>, Int. J. Algebra Comput. 9 (1999), 351-384. See also <a href="https://www.irif.fr/~cf/publications/fibgold.pdf">preprint</a>.
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/phigits.html">Phigits and the Base Phi representation</a>.
%H Ron Knott, <a href="/A105424/a105424.pdf">Phigits and the Base Phi representation</a> [Local copy, pdf only]
%H Jeffrey Shallit, <a href="https://arxiv.org/abs/2305.02672">Proving Properties of phi-Representations with the Walnut Theorem-Prover</a>, arXiv:2305.02672 [math.NT], 2023.
%e The first few numbers written in base phi are:
%e 0 = 0.
%e 1 = 1.
%e 2 = 10.01
%e 3 = 100.01
%e 4 = 101.01
%e 5 = 1000.1001
%e 6 = 1010.0001
%e 7 = 10000.0001
%e 8 = 10001.0001
%e 9 = 10010.0101
%e 10 = 10100.0101
%e 11 = 10101.0101
%e 12 = 100000.101001
%e 13 = 100010.001001
%e 14 = 100100.001001
%e 15 = 100101.001001
%e 16 = 101000.100001
%e 17 = 101010.000001
%e 18 = 1000000.000001
%e 19 = 1000001.000001
%e 20 = 1000010.010001
%e 21 = 1000100.010001
%e 22 = 1000101.010001
%e 23 = 1001000.100101
%e 24 = 1001010.000101
%e ...
%Y Cf. A105424, A105425.
%K nonn,base,easy
%O 0,3
%A _N. J. A. Sloane_, Mar 01 2021
%E Definition clarified by _N. J. A. Sloane_, May 27 2023