A256657 Numbers for which the minimal alternating Fibonacci representation has negative trace.

4, 6, 7, 10, 11, 12, 16, 18, 19, 20, 25, 26, 29, 31, 32, 33, 38, 40, 41, 42, 46, 47, 50, 52, 53, 54, 59, 61, 62, 65, 66, 67, 68, 72, 74, 75, 76, 80, 81, 84, 86, 87, 88, 93, 95, 96, 99, 100, 101, 105, 107, 108, 109, 110, 114, 116, 117, 120, 121, 122, 123, 127

Offset

1,1

Comments

See A256655 for definitions. This sequence and A256656 partition the positive integers.

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Example

Let R(k) be the minimal alternating Fibonacci representation of k.  The trace of R(k) is the last term.
R(1) = 1, trace = 1
R(2) = 2, trace = 2
R(3) = 3, trace = 3
R(4) = 5 - 1, trace = -1
R(5) = 5, trace = 5
R(6) = 6 - 2, trace = -2

Mathematica

b[n_] = Fibonacci[n]; bb = Table[b[n], {n, 1, 70}];
h[0] = {1}; h[n_] := Join[h[n - 1], Table[b[n + 2], {k, 1, b[n]}]];
g = h[18];  r[0] = {0};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]]
t = Table[Last[r[n]], {n, 0, 1000}];  (* A256656 *)
Select[Range[200], Last[r[#]] > 0 &]  (* A256656 *)
Select[Range[200], Last[r[#]] < 0 &]  (* A256657 *)

Crossrefs

Cf. A000045, A256655, A256656.

Sequence in context: A286494 A086170 A123088 * A189085 A153235 A286491

Adjacent sequences: A256654 A256655 A256656 * A256658 A256659 A256660

Keyword

nonn,easy

Author

Clark Kimberling, Apr 08 2015

Status

approved