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