A256656 Numbers for which the minimal alternating Fibonacci representation has positive trace.

1, 2, 3, 5, 8, 9, 13, 14, 15, 17, 21, 22, 23, 24, 27, 28, 30, 34, 35, 36, 37, 39, 43, 44, 45, 48, 49, 51, 55, 56, 57, 58, 60, 63, 64, 69, 70, 71, 73, 77, 78, 79, 82, 83, 85, 89, 90, 91, 92, 94, 97, 98, 102, 103, 104, 106, 111, 112, 113, 115, 118, 119, 124

Offset

1,2

Comments

See A256655 for definitions. This sequence and A256657 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, A256657.

Sequence in context: A162586 A232868 A189086 * A035422 A195896 A058237

Adjacent sequences: A256653 A256654 A256655 * A256657 A256658 A256659

Keyword

nonn,easy

Author

Clark Kimberling, Apr 08 2015

Status

approved