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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. 49
 1, 2, 5, 8, 26, 29, 50, 65, 68, 89, 128, 677, 680, 701, 740, 842, 845, 866, 905, 1352, 1517, 1682, 2501, 2504, 2525, 2564, 3176, 3341, 4226, 4229, 4250, 4289, 4625, 4628, 4649, 4688, 4901, 5000, 5066, 5300, 5465, 6725, 7124, 7922, 7925, 7946, 7985 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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: A192476:  f(x,y)=x^2+y^2, start={1} A003586:  f(x,y)=x*y, start={1,2,3} A051037:  f(x,y)=x*y, start={1,2,3,5} A002473:  f(x,y)=x*y, start={1,2,3,5,7} A003592:  f(x,y)=x*y, start={2,5} A009293:  f(x,y)=x*y+1, start={2} A009388:  f(x,y)=x*y-1, start={2} A009299:  f(x,y)=x*y+2, start={3} A192518:  f(x,y)=(x+1)(y+1), start={2} A192519:  f(x,y)=floor(x*y/2), start={3} A192520:  f(x,y)=floor(x*y/2), start={5} A192521:  f(x,y)=floor((x+1)(y+1)/2), start={2} A192522:  f(x,y)=floor((x-1)(y-1)/2), start={5} A192523:  f(x,y)=2x*y-x-y, start={2} A192525:  f(x,y)=2x*y-x-y, start={3} A192524:  f(x,y)=4x*y-x-y, start={1} A192528:  f(x,y)=5x*y-x-y, start={1} A192529:  f(x,y)=3x*y-x-y, start={2} A192531:  f(x,y)=3x*y-2x-2y, start={2} A192533:  f(x,y)=x^2+y^2-x*y, start={2} A192535:  f(x,y)=x^2+y^2+x*y, start={1} A192536:  f(x,y)=x^2+y^2-floor(x*y/2), start={1} A192537:  f(x,y)=x^2+y^2-x*y/2, start={2} A192539:  f(x,y)=2x*y+floor(x*y/2), start={1} LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..6171 EXAMPLE 1^2+1^2=2, 1^2+2^2=5, 2^2+2^2=8, 1^2+5^2=26. MATHEMATICA 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. *) 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 = FixedPoint[b, start] (* A192476 *) Differences[t] (* based on program by Robert G. Wilson v at A009293 *) PROG (Haskell) import Data.Set (singleton, deleteFindMin, insert) a192476 n = a192476_list !! (n-1) a192476_list = f  (singleton 1) where    f xs s =      m : f xs' (foldl (flip insert) s' (map (+ m^2) (map (^ 2) xs')))      where xs' = m : xs            (m, s') = deleteFindMin s -- Reinhard Zumkeller, Aug 15 2011 CROSSREFS Cf. A009293, A008318. Sequence in context: A291605 A142869 A086825 * A093365 A209865 A128600 Adjacent sequences:  A192473 A192474 A192475 * A192477 A192478 A192479 KEYWORD nonn AUTHOR Clark Kimberling, Jul 01 2011 STATUS approved

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Last modified July 28 10:51 EDT 2021. Contains 346326 sequences. (Running on oeis4.)