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

%I A256656 #4 Apr 09 2015 07:58:27
%S A256656 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,
%T A256656 48,49,51,55,56,57,58,60,63,64,69,70,71,73,77,78,79,82,83,85,89,90,91,
%U A256656 92,94,97,98,102,103,104,106,111,112,113,115,118,119,124
%N A256656 Numbers for which the minimal alternating Fibonacci representation has positive trace.
%C A256656 See A256655 for definitions.  This sequence and A256657 partition the positive integers.
%H A256656 Clark Kimberling, <a href="/A256656/b256656.txt">Table of n, a(n) for n = 1..1000</a>
%e A256656 Let R(k) be the minimal alternating Fibonacci representation of k.  The trace of R(k) is the last term.
%e A256656 R(1) = 1, trace = 1
%e A256656 R(2) = 2, trace = 2
%e A256656 R(3) = 3, trace = 3
%e A256656 R(4) = 5 - 1, trace = -1
%e A256656 R(5) = 5, trace = 5
%e A256656 R(6) = 6 - 2, trace =  -2
%t A256656 b[n_] = Fibonacci[n]; bb = Table[b[n], {n, 1, 70}];
%t A256656 h[0] = {1}; h[n_] := Join[h[n - 1], Table[b[n + 2], {k, 1, b[n]}]];
%t A256656 g = h[18];  r[0] = {0};
%t A256656 r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]]
%t A256656 t = Table[Last[r[n]], {n, 0, 1000}];  (* A256656 *)
%t A256656 Select[Range[200], Last[r[#]] > 0 &]  (* A256656 *)
%t A256656 Select[Range[200], Last[r[#]] < 0 &]  (* A256657 *)
%Y A256656 Cf. A000045, A256655, A256657.
%K A256656 nonn,easy
%O A256656 1,2
%A A256656 _Clark Kimberling_, Apr 08 2015