A198439 Square root of first term of a triple of squares in arithmetic progression that is not a multiple of another triple in (A198384, A198385, A198386).

1, 7, 7, 17, 1, 23, 31, 49, 17, 47, 23, 71, 7, 41, 41, 79, 97, 7, 31, 73, 127, 119, 89, 17, 161, 47, 113, 167, 119, 1, 199, 49, 73, 103, 161, 223, 241, 23, 31, 103, 89, 191, 287, 151, 217, 287, 137, 233, 71, 337, 79, 137, 17, 281, 359, 391, 49, 113, 119, 217

Offset

1,2

Comments

This sequence gives the values |x-y| of primitive Pythagorean triangles (x,y,z) with even y ordered according to the nondecreasing values of the leg sums x+y (called w in the Zumkeller link, and given in A198441). For the equivalence to primitive Pythagorean triples with even y see a comment in A198441. - Wolfdieter Lang, May 22 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

Formula

A198435(n) = a(n)^2; a(n) = A198388(A198409(n)).

Example

From Wolfdieter Lang, May 22 2013: (Start)
Primitive Pythagorean triple (x,y,z), y even, connection:
a(2) = 7 because the triple with second smallest leg sum x+y = 17 = A198441(2) is (5,12,13), and |x - y| = y - x = 12 - 5 = 7.
a(3) = 7 because x + y = A198441(3) = 23, (x,y,z) = (15,8,17) (the primitive triple with third smallest leg sum), and |x-y| = x - y = 15 - 8 = 7. (End)

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]];
tt = Flatten[DeleteCases[triples /@ Range[wmax], {}], 2];
DeleteCases[tt, t_List /; GCD@@t>1 && MemberQ[tt, t/GCD@@t]][[All, 1]] (* Jean-François Alcover, Oct 22 2021 *)

Prog

(Haskell)
a198439 n = a198439_list !! (n-1)
a198439_list = map a198388 a198409_list

Crossrefs

Sequence in context: A168411 A120682 A152910 * A100635 A292084 A168458

Adjacent sequences: A198436 A198437 A198438 * A198440 A198441 A198442

Keyword

nonn

Author

Reinhard Zumkeller, Oct 25 2011

Status

approved