login
Numbers for which the minimal alternating Fibonacci representation has positive trace.
6

%I #4 Apr 09 2015 07:58:27

%S 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 48,49,51,55,56,57,58,60,63,64,69,70,71,73,77,78,79,82,83,85,89,90,91,

%U 92,94,97,98,102,103,104,106,111,112,113,115,118,119,124

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

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

%H Clark Kimberling, <a href="/A256656/b256656.txt">Table of n, a(n) for n = 1..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[18]; r[0] = {0};

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

%t t = Table[Last[r[n]], {n, 0, 1000}]; (* A256656 *)

%t Select[Range[200], Last[r[#]] > 0 &] (* A256656 *)

%t Select[Range[200], Last[r[#]] < 0 &] (* A256657 *)

%Y Cf. A000045, A256655, A256657.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Apr 08 2015