A198389 Square root of second term of a triple of squares in arithmetic progression.

5, 10, 13, 15, 17, 20, 25, 26, 25, 29, 30, 34, 37, 35, 41, 39, 40, 50, 45, 52, 51, 50, 61, 53, 55, 65, 58, 60, 65, 65, 65, 68, 75, 74, 85, 70, 82, 78, 73, 75, 80, 85, 85, 85, 89, 91, 101, 87, 100, 90, 113, 95, 104, 97, 102, 100, 111, 122, 106, 105, 123, 109

Offset

1,1

Comments

Apart from its initial 1, A001653 is a subsequence: for all n>1 exists an m such that A198388(m)=1 and a(m)=A001653(n). [observed by Zak Seidov, Reinhard Zumkeller, Oct 25 2011]

There is a connection to hypotenuses of Pythagorean triangles. See a comment for the primitive case on A198441 which applies here mutatis mutandis. - Wolfdieter Lang, May 23 2013

Links

Ray Chandler, Table of n, a(n) for n = 1..10000

Reinhard Zumkeller, Table of initial values

Keith Conrad, Arithmetic progressions of three squares

Formula

A198385(n) = a(n)^2.

A198440(n) = a(A198409(n)).

Example

Connection to Pythagorean triangle hypotenuses: a(20) = 10 because (in the notation of the Zumkeller link) (u,v,w) = 2*(1,5,7) and the Pythagorean triangle is 2*(x=(7-1)/2,y=(1+7)/2,5) = 2*(3,4,5) with hypotenuse 2*5 = 10. - 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, 2]] (* Jean-François Alcover, Oct 20 2021 *)

Prog

(Haskell)
a198389 n = a198389_list !! (n-1)
a198389_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]

Crossrefs

Cf. A198385, A198409, A198440.

Sequence in context: A049197 A324928 A009000 * A057100 A304436 A009003

Adjacent sequences: A198386 A198387 A198388 * A198390 A198391 A198392

Keyword

nonn

Author

Reinhard Zumkeller, Oct 24 2011

Status

approved