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

1,1

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

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

Formula

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

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

Example

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

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 *)

Prog

(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]
(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

Crossrefs

Cf. A198386, A198409, A198439, A198441, A331671.

Sequence in context: A336797 A100599 A353440 * A118905 A254064 A257224

Adjacent sequences: A198387 A198388 A198389 * A198391 A198392 A198393

Keyword

nonn

Author

Reinhard Zumkeller, Oct 24 2011

Status

approved