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A353657 a(n) = A353655(n)- A353656(n).

%I A353657 #11 Dec 26 2024 21:43:38
%S A353657 0,-1,0,1,-1,0,2,-1,0,1,1,0,-1,0,0,0,-1,2,1,0,-1,-1,1,-1,-2,1,0,-2,2,
%T A353657 1,1,0,0,-1,-1,-1,0,-1,-1,0,-1,1,0,-1,-1,0,2,1,2,1,1,0,0,-1,-1,-1,-1,
%U A353657 -1,-1,1,0,-2,0,0,-1,-2,0,1,0,0,0,1,-2,-1,0,2,2
%N A353657 a(n) = A353655(n)- A353656(n).
%C A353657 Conjectures: a(n) = 0 for infinitely many n, and (a(n)) is unbounded below and above.
%e A353657 a(7) because A353655(u) = 3 and A353656(7) = 1, since the Fibonacci-Lucas representation of 7 is FL(7) = 5 + 1 + 1, and the Lucas-Fibonacci representation of 7 is LF(7) = 7.
%t A353657 z = 120; fib = Map[Fibonacci, Range[2, 51]];
%t A353657 luc = Map[LucasL, Range[1, 50]];
%t A353657 t = Map[(n = #; fl = {}; f = 0; l = 0;
%t A353657      While[IntegerQ[l], n = n - f - l;
%t A353657       f = fib[[NestWhile[# + 1 &, 1, fib[[#]] <= n &] - 1]];
%t A353657       l = luc[[NestWhile[# + 1 &, 1, luc[[#]] <= n - f &] - 1]];
%t A353657       AppendTo[fl, {f, l}]];
%t A353657      {Total[#], #} &[Select[Flatten[fl], IntegerQ]]) &, Range[z]];
%t A353657 u = Take[Map[Last, t], z];
%t A353657 u1 = Map[Length, u]  (* A353655 *)
%t A353657 t = Map[(n = #; lf = {}; f = 0; l = 0;
%t A353657      While[IntegerQ[f], n = n - l - f;
%t A353657       l = luc[[NestWhile[# + 1 &, 1, luc[[#]] <= n &] - 1]];
%t A353657       f = fib[[NestWhile[# + 1 &, 1, fib[[#]] <= n - l &] - 1]];
%t A353657       AppendTo[lf, {l, f}]];
%t A353657      {Total[#], #} &[Select[Flatten[lf], IntegerQ]]) &, Range[z]];
%t A353657 v = Take[Map[Last, t], z];
%t A353657 v1 = Map[Length, v]   (* A353656 *)
%t A353657 u1 - v1  (* A353657 *)
%t A353657 (* _Peter J. C. Moses_ *)
%Y A353657 Cf. A000032, A000045, A007895, A116543, A353655, A353656, A353658, A353659.
%K A353657 sign
%O A353657 1,7
%A A353657 _Clark Kimberling_, May 04 2022