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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 (list; graph; refs; listen; history; text; internal format)
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, June 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

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Last modified May 12 02:06 EDT 2021. Contains 343808 sequences. (Running on oeis4.)