A192645 Monotonic ordering of set S generated by these rules: if x and y are in S and x^2 - y^2 > 0 then x^2 - y^2 is in S, and 1 and 2 are in S.

1, 2, 3, 5, 8, 16, 21, 24, 39, 55, 60, 63, 135, 185, 192, 231, 247, 252, 255, 320, 369, 377, 416, 432, 437, 440, 512, 551, 567, 572, 575, 944, 945, 1080, 1265, 1457, 1496, 1504, 1512, 1517, 1520, 1521, 1889, 2079, 2448, 2449, 2495, 2584, 2631, 2639

Offset: 1

Author

Clark Kimberling, Jul 06 2011

Keywords

• nonn

Comments

See A192476 for a general discussion. Related sequences:

A192645: f(x,y) = x^2 - y^2 > 0, start={1,2};

A192647: f(x,y) = x^2 - y^2 > 0, start={1,3};

A192648: f(x,y) = x^2 - y^2 > 0, start={2,3};

A192649: f(x,y) = x^2 - y^2 > 0, start={1,2,4}.

Examples

			2^2 - 1^2 = 3;
3^2 - 2^2 = 5, 3^2 - 1^2 = 8;
5^2 - 3^2 = 16, 5^2 - 2^2 = 21, 5^2 - 1^2 = 24.
Taking the generating procedure in the order just indicated results in the monotonic ordering of the sequence and also suggests a triangular format for the generated terms:
    3;
    5,   8;
   16,  21,  24;
   39,  55,  60,  63;
  135, 185, 192, 231, 247;
  ...
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A192476, A192646 (first differences).

Programs

Mathematica

start = {1, 2};
f[x_, y_] := If[MemberQ[Range[1, 5000], x^2 - y^2], x^2 - y^2]
b[x_] :=
  Block[{w = x},
   Select[Union[
     Flatten[AppendTo[w,
       Table[f[w[[i]], w[[j]]], {i, 1, Length[w]}, {j, 1, i}]]]], # <
      5000 &]];
t = FixedPoint[b, start]  (* A192645 *)
Differences[t] (* A192646 *)