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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).

%I A105424 #37 Oct 20 2023 08:41:27
%S A105424 0,1,10,100,101,1000,1010,10000,10001,10010,10100,10101,100000,100010,
%T A105424 100100,100101,101000,101010,1000000,1000001,1000010,1000100,1000101,
%U A105424 1001000,1001010,1010000,1010001,1010010,1010100,1010101,10000000
%N 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).
%H A105424 T. D. Noe, <a href="/A105424/b105424.txt">Table of n, a(n) for n = 0..1000</a>
%H A105424 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 A105424 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 A105424 Ron Knott, <a href="/A105424/a105424.pdf">Phigits and the Base Phi representation</a> [Local copy, pdf only]
%H A105424 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. [Note that this document has been revised multiple times.]
%e A105424 2 = 10.01 in base phi, so left of the decimal point is 10.
%e A105424 The first few numbers written in base phi:
%e A105424 0 = 0.
%e A105424 1 = 1.
%e A105424 2 = 10.01
%e A105424 3 = 100.01
%e A105424 4 = 101.01
%e A105424 5 = 1000.1001
%e A105424 6 = 1010.0001
%e A105424 7 = 10000.0001
%e A105424 8 = 10001.0001
%e A105424 9 = 10010.0101
%e A105424 10 = 10100.0101
%e A105424 11 = 10101.0101
%e A105424 12 = 100000.101001
%e A105424 13 = 100010.001001
%e A105424 14 = 100100.001001
%e A105424 15 = 100101.001001
%e A105424 16 = 101000.100001
%e A105424 17 = 101010.000001
%e A105424 18 = 1000000.000001
%e A105424 19 = 1000001.000001
%e A105424 20 = 1000010.010001
%e A105424 21 = 1000100.010001
%e A105424 22 = 1000101.010001
%e A105424 23 = 1001000.100101
%e A105424 24 = 1001010.000101
%e A105424 ...
%t A105424 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 *)
%Y A105424 Cf. A001622, A055778, A105425, A104605.
%Y A105424 See A341722 for the part to the right of the decimal point.
%Y A105424 Cf. A105116 (base e), A344939 (base Pi).
%K A105424 nonn,base
%O A105424 0,3
%A A105424 Bryan Jacobs (bryanjj(AT)gmail.com), Apr 08 2005
%E A105424 Definition clarified by _N. J. A. Sloane_, May 27 2023