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