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

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, 12, 35, 23, 13, 28, 51, 46, 71, 14, 62, 42, 7, 15, 16, 41, 35, 17, 41, 49, 79, 3, 68, 18, 97, 19, 56, 7, 42, 20, 69, 98, 34, 21, 93, 31, 63, 22, 85, 94, 23, 49, 73

Offset: 1

Author

Reinhard Zumkeller, Oct 24 2011

Keywords

• nonn

Comments

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

Examples

			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
		

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000
• Keith Conrad, Arithmetic progressions of three squares
• Reinhard Zumkeller, Table of initial values

Crossrefs

Cf. A198384, A198409, A198439.

Programs

Haskell

a198388 n = a198388_list !! (n-1)
a198388_list = map (\(x,,) -> x) ts where
   ts = [(u,v,w) | w <- [1..], v <- [1..w-1], u <- [1..v-1],
                   w^2 - v^2 == v^2 - u^2]

Mathematica

wmax = 1000;
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]];
Flatten[DeleteCases[triples /@ Range[wmax], {}], 2][[All, 1]] (* Jean-François Alcover, Oct 20 2021 *)

Formula

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

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