%I #5 Apr 09 2015 07:58:00

%S 0,1,2,3,-1,5,-2,-1,8,1,-3,-2,-1,13,1,2,-5,1,-3,-2,-1,21,1,2,3,-1,-8,

%T 1,2,-5,1,-3,-2,-1,34,1,2,3,-1,5,-2,-1,-13,1,2,3,-1,-8,1,2,-5,1,-3,-2,

%U -1,55,1,2,3,-1,5,-2,-1,8,1,-3,-2,-1,-21,1,2,3,-1

%N Trace of the minimal alternating Fibonacci representation of n.

%C See A256655 for definitions.

%H Clark Kimberling, <a href="/A255973/b255973.txt">Table of n, a(n) for n = 0..1000</a>

%e Let R(k) be the minimal alternating Fibonacci representation of k. The trace of R(k) is the last term.

%e R(1) = 1, trace = 1

%e R(2) = 2, trace = 2

%e R(3) = 3, trace = 3

%e R(4) = 5 - 1, trace = -1

%e R(5) = 5, trace = 5

%e R(6) = 6 - 2, trace = -2

%t b[n_] = Fibonacci[n]; bb = Table[b[n], {n, 1, 70}];

%t h[0] = {1}; h[n_] := Join[h[n - 1], Table[b[n + 2], {k, 1, b[n]}]];

%t g = h[12]; r[0] = {0};

%t r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]];

%t Table[Last[r[n]], {n, 0, 200}] (* A255973 *)

%Y Cf. A000045, A256655 (representations R(n)), A256656 (numbers with positive trace), A256657 (numbers with nonpositive trace), A256663 (positive part of R(n)), A256664 (nonpositive part of R(n)), A256654.

%K easy,sign

%O 0,3

%A _Clark Kimberling_, Apr 08 2015