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A336007 Numbers whose mixed Zeckendorf-Lucas representation is not a Zeckendorf or Lucas representation. See Comments.

%I A336007 #7 Feb 06 2022 14:43:05
%S A336007 17,25,28,38,41,45,46,52,53,59,62,66,67,72,73,74,75,81,82,84,85,86,93,
%T A336007 96,100,101,106,107,108,109,114,117,118,119,120,121,122,128,129,131,
%U A336007 132,133,136,137,138,139,140,148,151,155,156,161,162,163,164,169
%N A336007 Numbers whose mixed Zeckendorf-Lucas representation is not a Zeckendorf or Lucas representation. See Comments.
%C A336007 Suppose that B1 and B2 are increasing sequences of positive integers, and let B be the increasing sequence of numbers in the union of B1 and B2. Every positive integer n has a unique representation given by the greedy algorithm with B1 as base, and likewise for B2 and B.
%e A336007 17 = 13 + 4;
%e A336007 25 = 21 + 4;
%e A336007 28 = 21 + 7.
%t A336007 fibonacciQ[n_] := IntegerQ[Sqrt[5 n^2 + 4]] || IntegerQ[Sqrt[5 n^2 - 4]];
%t A336007 Attributes[fibonacciQ] = {Listable};
%t A336007 lucasQ[n_] := IntegerQ[Sqrt[5 n^2 + 20]] || IntegerQ[Sqrt[5 n^2 - 20]];
%t A336007 Attributes[lucasQ] = {Listable};
%t A336007 s = Reverse[Union[Flatten[Table[{Fibonacci[n + 1], LucasL[n - 1]}, {n, 1, 22}]]]];
%t A336007 u = Map[#[[1]] &, Select[Map[{#[[1]], {Apply[And, fibonacciQ[#[[2]]]],
%t A336007        Apply[And, lucasQ[#[[2]]]]}} &, Map[{#, DeleteCases[
%t A336007         s Reap[FoldList[(Sow[Quotient[#1, #2]]; Mod[#1, #2]) &, #,
%t A336007             s]][[2, 1]], 0]} &,
%t A336007      Range[500]]], #[[2]] == {False, False} &]]
%t A336007 (* _Peter J. C. Moses_, Jun 14 2020 *)
%Y A336007 Cf. A007895, A014417, A116543, A214973, A336004.
%K A336007 nonn,base
%O A336007 1,1
%A A336007 _Clark Kimberling_, Jul 06 2020