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