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

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, 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, -1, -1, 1, 0, -2, 0, 0, -1, -2, 0, 1, 0, 0, 0, 1, -2, -1, 0, 2, 2

Offset: 1

Author

Clark Kimberling, May 04 2022

Keywords

• sign

Comments

Conjectures: a(n) = 0 for infinitely many n, and (a(n)) is unbounded below and above.

Examples

			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.
		

Crossrefs

Cf. A000032, A000045, A007895, A116543, A353655, A353656, A353658, A353659.

Programs

Mathematica

z = 120; fib = Map[Fibonacci, Range[2, 51]];
luc = Map[LucasL, Range[1, 50]];
t = Map[(n = #; fl = {}; f = 0; l = 0;
     While[IntegerQ[l], n = n - f - l;
      f = fib[[NestWhile[# + 1 &, 1, fib[[#]] <= n &] - 1]];
      l = luc[[NestWhile[# + 1 &, 1, luc[[#]] <= n - f &] - 1]];
      AppendTo[fl, {f, l}]];
     {Total[#], #} &[Select[Flatten[fl], IntegerQ]]) &, Range[z]];
u = Take[Map[Last, t], z];
u1 = Map[Length, u]  (* A353655 *)
t = Map[(n = #; lf = {}; f = 0; l = 0;
     While[IntegerQ[f], n = n - l - f;
      l = luc[[NestWhile[# + 1 &, 1, luc[[#]] <= n &] - 1]];
      f = fib[[NestWhile[# + 1 &, 1, fib[[#]] <= n - l &] - 1]];
      AppendTo[lf, {l, f}]];
     {Total[#], #} &[Select[Flatten[lf], IntegerQ]]) &, Range[z]];
v = Take[Map[Last, t], z];
v1 = Map[Length, v]   (* A353656 *)
u1 - v1  (* A353657 *)
(* Peter J. C. Moses *)