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a(n) = A353655(n)- A353656(n).
3

%I #11 Dec 26 2024 21:43:38

%S 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 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 -1,-1,1,0,-2,0,0,-1,-2,0,1,0,0,0,1,-2,-1,0,2,2

%N a(n) = A353655(n)- A353656(n).

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

%e 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 z = 120; fib = Map[Fibonacci, Range[2, 51]];

%t luc = Map[LucasL, Range[1, 50]];

%t t = Map[(n = #; fl = {}; f = 0; l = 0;

%t While[IntegerQ[l], n = n - f - l;

%t f = fib[[NestWhile[# + 1 &, 1, fib[[#]] <= n &] - 1]];

%t l = luc[[NestWhile[# + 1 &, 1, luc[[#]] <= n - f &] - 1]];

%t AppendTo[fl, {f, l}]];

%t {Total[#], #} &[Select[Flatten[fl], IntegerQ]]) &, Range[z]];

%t u = Take[Map[Last, t], z];

%t u1 = Map[Length, u] (* A353655 *)

%t t = Map[(n = #; lf = {}; f = 0; l = 0;

%t While[IntegerQ[f], n = n - l - f;

%t l = luc[[NestWhile[# + 1 &, 1, luc[[#]] <= n &] - 1]];

%t f = fib[[NestWhile[# + 1 &, 1, fib[[#]] <= n - l &] - 1]];

%t AppendTo[lf, {l, f}]];

%t {Total[#], #} &[Select[Flatten[lf], IntegerQ]]) &, Range[z]];

%t v = Take[Map[Last, t], z];

%t v1 = Map[Length, v] (* A353656 *)

%t u1 - v1 (* A353657 *)

%t (* _Peter J. C. Moses_ *)

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

%K sign

%O 1,7

%A _Clark Kimberling_, May 04 2022