

A198390


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


8



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
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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 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


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]] (* JeanFrançois Alcover, Oct 20 2021 *)


PROG

(Haskell)
a198390 n = a198390_list !! (n1)
a198390_list = map (\(_, _, x) > x) ts where
ts = [(u, v, w)  w < [1..], v < [1..w1], u < [1..v1],
w^2  v^2 == v^2  u^2]
(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


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



