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.

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

Offset: 1

Author

Clark Kimberling, Jul 01 2011

Keywords

• nonn

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}

Examples

			1^2+1^2=2, 1^2+2^2=5, 2^2+2^2=8, 1^2+5^2=26.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..6171

Crossrefs

Cf. A009293, A008318.

Programs

Haskell

import Data.Set (singleton, deleteFindMin, insert)
a192476 n = a192476_list !! (n-1)
a192476_list = f [1] (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

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 *)