A198388 - OEIS

%I #30 Oct 20 2021 10:21:34

%S 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 12,35,23,13,28,51,46,71,14,62,42,7,15,16,41,35,17,41,49,79,3,68,18,

%U 97,19,56,7,42,20,69,98,34,21,93,31,63,22,85,94,23,49,73

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

%C 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 Ray Chandler, <a href="/A198388/b198388.txt">Table of n, a(n) for n = 1..10000</a>

%H Keith Conrad, <a href="http://www.math.uconn.edu/~kconrad/blurbs/ugradnumthy/3squarearithprog.pdf">Arithmetic progressions of three squares</a>

%H Reinhard Zumkeller, <a href="/A198384/a198384_2.txt">Table of initial values</a>

%F A198384(n) = a(n)^2.

%F A198439(n) = a(A198409(n)).

%e 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 wmax = 1000;

%t 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 Flatten[DeleteCases[triples /@ Range[wmax], {}], 2][[All, 1]] (* _Jean-François Alcover_, Oct 20 2021 *)

%o (Haskell)

%o a198388 n = a198388_list !! (n-1)

%o a198388_list = map (\(x,_,_) -> x) ts where

%o    ts = [(u,v,w) | w <- [1..], v <- [1..w-1], u <- [1..v-1],

%o                    w^2 - v^2 == v^2 - u^2]

%Y Cf. A198384, A198409, A198439.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Oct 24 2011