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A130609
Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+223)^2 = y^2.
6
0, 32, 533, 669, 833, 3672, 4460, 5412, 21945, 26537, 32085, 128444, 155208, 187544, 749165, 905157, 1093625, 4366992, 5276180, 6374652, 25453233, 30752369, 37154733, 148352852, 179238480, 216554192, 864664325, 1044678957, 1262170865, 5039633544, 6088835708
OFFSET
1,2
COMMENTS
Also values x of Pythagorean triples (x, x+223, y).
Corresponding values y of solutions (x, y) are in A159809.
For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number > 7 in A028871, see A118337.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (227+30*sqrt(2))/223 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (105507+65798*sqrt(2))/223^2 for n mod 3 = 0.
FORMULA
a(n) = 6*a(n-3)-a(n-6)+446 for n > 6; a(1)=0, a(2)=32, a(3)=533, a(4)=669, a(5)=833, a(6)=3672.
G.f.: x*(32+501*x+136*x^2-28*x^3-167*x^4-28*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 223*A001652(k) for k >= 0.
MATHEMATICA
LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 32, 533, 669, 833, 3672, 4460}, 70] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2012 *)
PROG
(PARI) {forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+446*n+49729), print1(n, ", ")))}
CROSSREFS
Cf. A159809, A028871, A118337, A118675, A118676, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159810 (decimal expansion of (227+30*sqrt(2))/223), A159811 (decimal expansion of (105507+65798*sqrt(2))/223^2).
Sequence in context: A162739 A010984 A022596 * A154306 A004417 A283688
KEYWORD
nonn,easy
AUTHOR
Mohamed Bouhamida, Jun 17 2007
EXTENSIONS
Edited and two terms added by Klaus Brockhaus, Apr 30 2009
STATUS
approved