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A353657
a(n) = A353655(n)- A353656(n).
3
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
1,7
COMMENTS
Conjectures: a(n) = 0 for infinitely many n, and (a(n)) is unbounded below and above.
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;
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 *)
KEYWORD
sign
AUTHOR
Clark Kimberling, May 04 2022
STATUS
approved