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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.

%I A192645 #15 Sep 14 2018 09:56:44
%S A192645 1,2,3,5,8,16,21,24,39,55,60,63,135,185,192,231,247,252,255,320,369,
%T A192645 377,416,432,437,440,512,551,567,572,575,944,945,1080,1265,1457,1496,
%U A192645 1504,1512,1517,1520,1521,1889,2079,2448,2449,2495,2584,2631,2639
%N 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.
%C A192645 See A192476 for a general discussion.  Related sequences:
%C A192645 A192645:  f(x,y) = x^2 - y^2 > 0, start={1,2};
%C A192645 A192647:  f(x,y) = x^2 - y^2 > 0, start={1,3};
%C A192645 A192648:  f(x,y) = x^2 - y^2 > 0, start={2,3};
%C A192645 A192649:  f(x,y) = x^2 - y^2 > 0, start={1,2,4}.
%H A192645 Ivan Neretin, <a href="/A192645/b192645.txt">Table of n, a(n) for n = 1..10000</a>
%e A192645 2^2 - 1^2 = 3;
%e A192645 3^2 - 2^2 = 5, 3^2 - 1^2 = 8;
%e A192645 5^2 - 3^2 = 16, 5^2 - 2^2 = 21, 5^2 - 1^2 = 24.
%e A192645 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:
%e A192645     3;
%e A192645     5,   8;
%e A192645    16,  21,  24;
%e A192645    39,  55,  60,  63;
%e A192645   135, 185, 192, 231, 247;
%e A192645   ...
%t A192645 start = {1, 2};
%t A192645 f[x_, y_] := If[MemberQ[Range[1, 5000], x^2 - y^2], x^2 - y^2]
%t A192645 b[x_] :=
%t A192645   Block[{w = x},
%t A192645    Select[Union[
%t A192645      Flatten[AppendTo[w,
%t A192645        Table[f[w[[i]], w[[j]]], {i, 1, Length[w]}, {j, 1, i}]]]], # <
%t A192645       5000 &]];
%t A192645 t = FixedPoint[b, start]  (* A192645 *)
%t A192645 Differences[t] (* A192646 *)
%Y A192645 Cf. A192476, A192646 (first differences).
%K A192645 nonn
%O A192645 1,2
%A A192645 _Clark Kimberling_, Jul 06 2011