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A198388 Square root of first term of a triple of squares in arithmetic progression.

%I A198388 #30 Oct 20 2021 10:21:34
%S A198388 1,2,7,3,7,4,17,14,5,1,6,14,23,7,31,21,8,34,9,28,21,10,49,17,11,47,2,
%T A198388 12,35,23,13,28,51,46,71,14,62,42,7,15,16,41,35,17,41,49,79,3,68,18,
%U A198388 97,19,56,7,42,20,69,98,34,21,93,31,63,22,85,94,23,49,73
%N A198388 Square root of first term of a triple of squares in arithmetic progression.
%C A198388 There is a connection to |x-y| of Pythagorean triangles (x,y,z). See a comment on the primitive Pythagorean triangle case under A198441 which applies mutatis mutandis. - _Wolfdieter Lang_, May 23 2013
%H A198388 Ray Chandler, <a href="/A198388/b198388.txt">Table of n, a(n) for n = 1..10000</a>
%H A198388 Keith Conrad, <a href="http://www.math.uconn.edu/~kconrad/blurbs/ugradnumthy/3squarearithprog.pdf">Arithmetic progressions of three squares</a>
%H A198388 Reinhard Zumkeller, <a href="/A198384/a198384_2.txt">Table of initial values</a>
%F A198388 A198384(n) = a(n)^2.
%F A198388 A198439(n) = a(A198409(n)).
%e A198388 Connection to Pythagorean triangles: a(2) = 2 because (in the notation of the Zumkeller link) (u,v,w) = 2*(1,5,7) and the corresponding Pythagorean triangle is 2*((7-1)/2,(1+7)/2,5) = 2*(3,4,5) with |x-y| = 2*(4-3) = 2. - _Wolfdieter Lang_, May 23 2013
%t A198388 wmax = 1000;
%t A198388 triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u, v, w}]]]]][[2]];
%t A198388 Flatten[DeleteCases[triples /@ Range[wmax], {}], 2][[All, 1]] (* _Jean-François Alcover_, Oct 20 2021 *)
%o A198388 (Haskell)
%o A198388 a198388 n = a198388_list !! (n-1)
%o A198388 a198388_list = map (\(x,_,_) -> x) ts where
%o A198388    ts = [(u,v,w) | w <- [1..], v <- [1..w-1], u <- [1..v-1],
%o A198388                    w^2 - v^2 == v^2 - u^2]
%Y A198388 Cf. A198384, A198409, A198439.
%K A198388 nonn
%O A198388 1,2
%A A198388 _Reinhard Zumkeller_, Oct 24 2011