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

1,1

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.

Links

Table of n, a(n) for n=1..56.

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

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

Sequence in context: A124971 A272635 A105448 * A082130 A140609 A131275

Adjacent sequences: A336004 A336005 A336006 * A336008 A336009 A336010

Keyword

nonn,base

Author

Clark Kimberling, Jul 06 2020

Status

approved