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A192476 Monotonic ordering of set S generated by these rules: if x and y are in S then x^2 + y^2 is in S, and 1 is in S.

%I A192476 #15 Dec 18 2015 18:14:47
%S A192476 1,2,5,8,26,29,50,65,68,89,128,677,680,701,740,842,845,866,905,1352,
%T A192476 1517,1682,2501,2504,2525,2564,3176,3341,4226,4229,4250,4289,4625,
%U A192476 4628,4649,4688,4901,5000,5066,5300,5465,6725,7124,7922,7925,7946,7985
%N A192476 Monotonic ordering of set S generated by these rules: if x and y are in S then x^2 + y^2 is in S, and 1 is in S.
%C A192476 Let N denote the positive integers, and suppose that f(x,y): N x N->N. Let "start" denote a subset of N, and let S be the set of numbers defined by these rules: if x and y are in S, then f(x,y) is in S, and "start" is a subset of S. The monotonic increasing ordering of S is a sequence:
%C A192476 A192476:  f(x,y)=x^2+y^2, start={1}
%C A192476 A003586:  f(x,y)=x*y, start={1,2,3}
%C A192476 A051037:  f(x,y)=x*y, start={1,2,3,5}
%C A192476 A002473:  f(x,y)=x*y, start={1,2,3,5,7}
%C A192476 A003592:  f(x,y)=x*y, start={2,5}
%C A192476 A009293:  f(x,y)=x*y+1, start={2}
%C A192476 A009388:  f(x,y)=x*y-1, start={2}
%C A192476 A009299:  f(x,y)=x*y+2, start={3}
%C A192476 A192518:  f(x,y)=(x+1)(y+1), start={2}
%C A192476 A192519:  f(x,y)=floor(x*y/2), start={3}
%C A192476 A192520:  f(x,y)=floor(x*y/2), start={5}
%C A192476 A192521:  f(x,y)=floor((x+1)(y+1)/2), start={2}
%C A192476 A192522:  f(x,y)=floor((x-1)(y-1)/2), start={5}
%C A192476 A192523:  f(x,y)=2x*y-x-y, start={2}
%C A192476 A192525:  f(x,y)=2x*y-x-y, start={3}
%C A192476 A192524:  f(x,y)=4x*y-x-y, start={1}
%C A192476 A192528:  f(x,y)=5x*y-x-y, start={1}
%C A192476 A192529:  f(x,y)=3x*y-x-y, start={2}
%C A192476 A192531:  f(x,y)=3x*y-2x-2y, start={2}
%C A192476 A192533:  f(x,y)=x^2+y^2-x*y, start={2}
%C A192476 A192535:  f(x,y)=x^2+y^2+x*y, start={1}
%C A192476 A192536:  f(x,y)=x^2+y^2-floor(x*y/2), start={1}
%C A192476 A192537:  f(x,y)=x^2+y^2-x*y/2, start={2}
%C A192476 A192539:  f(x,y)=2x*y+floor(x*y/2), start={1}
%H A192476 Reinhard Zumkeller, <a href="/A192476/b192476.txt">Table of n, a(n) for n = 1..6171</a>
%e A192476 1^2+1^2=2, 1^2+2^2=5, 2^2+2^2=8, 1^2+5^2=26.
%t A192476 start = {1}; f[x_, y_] :=  x^2 + y^2  (* start is a subset of t, and if x,y are in t then f(x,y) is in t. *)
%t A192476 b[z_] :=  Block[{w = z}, Select[Union[Flatten[AppendTo[w, Table[f[w[[i]], w[[j]]], {i, 1, Length[w]}, {j, 1, i}]]]], # < 30000 &]];
%t A192476 t = FixedPoint[b, start] (* A192476 *)
%t A192476 Differences[t]
%t A192476 (* based on program by _Robert G. Wilson v_ at A009293 *)
%o A192476 (Haskell)
%o A192476 import Data.Set (singleton, deleteFindMin, insert)
%o A192476 a192476 n = a192476_list !! (n-1)
%o A192476 a192476_list = f [1] (singleton 1) where
%o A192476    f xs s =
%o A192476      m : f xs' (foldl (flip insert) s' (map (+ m^2) (map (^ 2) xs')))
%o A192476      where xs' = m : xs
%o A192476            (m,s') = deleteFindMin s
%o A192476 -- _Reinhard Zumkeller_, Aug 15 2011
%Y A192476 Cf. A009293, A008318.
%K A192476 nonn
%O A192476 1,2
%A A192476 _Clark Kimberling_, Jul 01 2011