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

1,2

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}.

Links

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

Example

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;
  ...

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 *)

Crossrefs

Cf. A192476, A192646 (first differences).

Sequence in context: A305370 A192571 A215525 * A050295 A121649 A306622

Adjacent sequences: A192642 A192643 A192644 * A192646 A192647 A192648

Keyword

nonn

Author

Clark Kimberling, Jul 06 2011

Status

approved