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A362919 a(n) is the right portion (reversed) of the base-phi representation of n in Knott's representation which uses the least number of 0's, the most 1's, and in which the right-hand portion is finite.

%I A362919 #25 May 27 2023 15:08:37
%S A362919 0,0,11,1111,1111,1111,1110,111101,111101,111101,111111,111111,111111,
%T A362919 111110,111011,111011,111011,111010,11110101,11110101,11110101,
%U A362919 11110111,11110111,11110111,11110110,11111101,11111101,11111101,11111111,11111111,11111111
%N A362919 a(n) is the right portion (reversed) of the base-phi representation of n in Knott's representation which uses the least number of 0's, the most 1's, and in which the right-hand portion is finite.
%C A362919 The left portion is given in A118240.
%H A362919 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 A362919 Ron Knott, <a href="/A105424/a105424.pdf">Phigits and the Base Phi representation</a> [Local copy, pdf only]
%H A362919 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 A362919 The representations of the numbers 0 though 30 are:
%e A362919   0 = 0.0
%e A362919   1 = 1.0
%e A362919   2 = 1.11
%e A362919   3 = 10.1111
%e A362919   4 = 11.1111
%e A362919   5 = 101.1111
%e A362919   6 = 111.0111
%e A362919   7 = 1010.101111
%e A362919   8 = 1011.101111
%e A362919   9 = 1101.101111
%e A362919   10 = 1110.111111
%e A362919   11 = 1111.111111
%e A362919   12 = 10101.111111
%e A362919   13 = 10111.011111
%e A362919   14 = 11010.110111
%e A362919   15 = 11011.110111
%e A362919   16 = 11101.110111
%e A362919   17 = 11111.010111
%e A362919   18 = 101010.10101111
%e A362919   19 = 101011.10101111
%e A362919   20 = 101101.10101111
%e A362919   21 = 101110.11101111
%e A362919   22 = 101111.11101111
%e A362919   23 = 110101.11101111
%e A362919   24 = 110111.01101111
%e A362919   25 = 111010.10111111
%e A362919   26 = 111011.10111111
%e A362919   27 = 111101.10111111
%e A362919   28 = 111110.11111111
%e A362919   29 = 111111.11111111
%e A362919   30 = 1010101.11111111
%Y A362919 Cf. A118240, A362920.
%K A362919 nonn,base
%O A362919 0,3
%A A362919 _N. J. A. Sloane_, May 27 2023