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

17, 25, 28, 38, 41, 45, 46, 52, 53, 59, 62, 66, 67, 72, 73, 74, 75, 81, 82, 84, 85, 86, 93, 96, 100, 101, 106, 107, 108, 109, 114, 117, 118, 119, 120, 121, 122, 128, 129, 131, 132, 133, 136, 137, 138, 139, 140, 148, 151, 155, 156, 161, 162, 163, 164, 169

Offset: 1

Author

Clark Kimberling, Jul 06 2020

Keywords

• nonn
• base

Comments

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.

Examples

			17 = 13 + 4;
25 = 21 + 4;
28 = 21 + 7.
		

Crossrefs

Cf. A007895, A014417, A116543, A214973, A336004.

Programs

Mathematica

fibonacciQ[n_] := IntegerQ[Sqrt[5 n^2 + 4]] || IntegerQ[Sqrt[5 n^2 - 4]];
Attributes[fibonacciQ] = {Listable};
lucasQ[n_] := IntegerQ[Sqrt[5 n^2 + 20]] || IntegerQ[Sqrt[5 n^2 - 20]];
Attributes[lucasQ] = {Listable};
s = Reverse[Union[Flatten[Table[{Fibonacci[n + 1], LucasL[n - 1]}, {n, 1, 22}]]]];
u = Map[#[[1]] &, Select[Map[{#[[1]], {Apply[And, fibonacciQ[#[[2]]]],
       Apply[And, lucasQ[#[[2]]]]}} &, Map[{#, DeleteCases[
        s Reap[FoldList[(Sow[Quotient[#1, #2]]; Mod[#1, #2]) &, #,
            s]][[2, 1]], 0]} &,
     Range[500]]], #[[2]] == {False, False} &]]
(* Peter J. C. Moses, Jun 14 2020 *)