A353657 - OEIS

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

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OFFSET

1,7

COMMENTS

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

LINKS

Table of n, a(n) for n=1..77.

EXAMPLE

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.

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;