%I
%S 7,14,17,21,23,28,31,34,35,41,42,46,47,49,49,51,56,62,63,68,69,70,71,
%T 73,77,79,82,84,85,89,91,92,93,94,97,98,98,102,103,105,112,113,115,
%U 119,119,119,119,123,124,126,127,133,136,137,138,140,141,142,146
%N Square root of third term of a triple of squares in arithmetic progression.
%C A198386(n) = a(n)^2.
%C A198441(n) = a(A198409(n)).
%C There is a connection to the leg sums of Pythagorean triangles.
%C See a comment on the primitive case under A198439, which applies mutatis mutandis.  _Wolfdieter Lang_, May 23 2013
%C Are these just the positive multiples of A001132?  _Charles R Greathouse IV_, May 28 2013
%C n appears A331671(n) times.  _Ray Chandler_, Feb 26 2020
%H Ray Chandler, <a href="/A198390/b198390.txt">Table of n, a(n) for n = 1..10000</a>
%H Reinhard Zumkeller, <a href="/A198384/a198384_2.txt">Table of initial values</a>
%H Keith Conrad, <a href="http://www.math.uconn.edu/~kconrad/blurbs/ugradnumthy/3squarearithprog.pdf">Arithmetic progressions of three squares</a>
%e 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 nonprimitive Pythagorean triangle 2*(x=(71)/1,y=(1+7)/2,z=5) = 2*(3,4,5) with leg sum 2*(3+4) = 14.  _Wolfdieter Lang_, May 23 2013
%o (Haskell)
%o a198390 n = a198390_list !! (n1)
%o a198390_list = map (\(_,_,x) > x) ts where
%o ts = [(u,v,w)  w < [1..], v < [1..w1], u < [1..v1],
%o w^2  v^2 == v^2  u^2]
%o (PARI) is(n)=my(t=n^2);forstep(i=2n%2,n2,2, if(issquare((t+i^2)/2), return(1))); 0 \\ _Charles R Greathouse IV_, May 28 2013
%Y Cf. A198386, A198409, A198439, A198441, A331671.
%K nonn
%O 1,1
%A _Reinhard Zumkeller_, Oct 24 2011
