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

Author

Clark Kimberling, Apr 08 2015

Keywords

• nonn
• easy

Comments

See A256655 for definitions. This sequence and A256657 partition the positive integers.

Examples

			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
		

Links

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

Crossrefs

Cf. A000045, A256655, A256657.

Programs

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 *)