A198390 Square root of third term of a triple of squares in arithmetic progression.

7, 14, 17, 21, 23, 28, 31, 34, 35, 41, 42, 46, 47, 49, 49, 51, 56, 62, 63, 68, 69, 70, 71, 73, 77, 79, 82, 84, 85, 89, 91, 92, 93, 94, 97, 98, 98, 102, 103, 105, 112, 113, 115, 119, 119, 119, 119, 123, 124, 126, 127, 133, 136, 137, 138, 140, 141, 142, 146

Offset: 1

Author

Reinhard Zumkeller, Oct 24 2011

Keywords

• nonn

Comments

There is a connection to the leg sums of Pythagorean triangles.

See a comment on the primitive case under A198439, which applies mutatis mutandis. - Wolfdieter Lang, May 23 2013

Are these just the positive multiples of A001132? - Charles R Greathouse IV, May 28 2013

n appears A331671(n) times. - Ray Chandler, Feb 26 2020

Examples

			Connection to leg sums of Pythagorean triangles: a(2) = 14 because (in the notation of the Zumkeller link) (u,v,w)= (2,10,14) = 2*(1,5,7), and this corresponds to the non-primitive Pythagorean triangle 2*(x=(7-1)/1,y=(1+7)/2,z=5) = 2*(3,4,5) with leg sum 2*(3+4) = 14. - _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. A198386, A198409, A198439, A198441, A331671.

Programs

Haskell

a198390 n = a198390_list !! (n-1)
a198390_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, 3]] (* Jean-François Alcover, Oct 20 2021 *)

PARI

is(n)=my(t=n^2);forstep(i=2-n%2,n-2,2, if(issquare((t+i^2)/2), return(1))); 0 \\ Charles R Greathouse IV, May 28 2013

Formula

A198386(n) = a(n)^2.

A198441(n) = a(A198409(n)).