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Numbers whose minimal alternating squares representation has positive trace.
4

%I #6 Apr 14 2015 11:04:59

%S 2,3,5,8,10,12,15,18,19,21,24,27,29,30,32,35,38,40,42,43,45,48,50,51,

%T 53,55,57,58,60,63,65,67,68,70,72,74,75,77,80,83,84,86,87,89,91,93,94,

%U 96,99,101,104,105,107,108,110,112,114,115,117,120,122,124

%N Numbers whose minimal alternating squares representation has positive trace.

%C See A256789 for definitions.

%H Clark Kimberling, <a href="/A256793/b256793.txt">Table of n, a(n) for n = 1..1000</a>

%e R(1) = 1; trace = 1, positive.

%e R(2) = 4 - 2; trace = -2, negative.

%e R(3) = 4 - 1; trace = -1, negative.

%t b[n_] := n^2; bb = Table[b[n], {n, 0, 1000}];

%t s[n_] := Table[b[n], {k, 1, 2 n - 1}];

%t h[1] = {1}; h[n_] := Join[h[n - 1], s[n]];

%t g = h[100]; r[0] = {0}; r[1] = {1}; r[2] = {4, -2};

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

%t Table[r[n], {n, 0, 120}] (* A256789 *)

%t u = Flatten[Table[Last[r[n]], {n, 1, 1000}]]; (* A256791 *)

%t Select[Range[800], u[[#]] > 0 &] (* A256792 *)

%t Select[Range[800], u[[#]] < 0 &] (* A256793 *)

%Y Cf. A256789, A256792 (complement).

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Apr 13 2015