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A336007
Numbers whose mixed Zeckendorf-Lucas representation is not a Zeckendorf or Lucas representation. See Comments.
1
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
1,1
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.
EXAMPLE
17 = 13 + 4;
25 = 21 + 4;
28 = 21 + 7.
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 *)
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Clark Kimberling, Jul 06 2020
STATUS
approved