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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).
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%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