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A123654
Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+809)^2 = y^2.
6
0, 264, 1491, 2427, 3811, 10764, 16180, 24220, 64711, 96271, 143127, 379120, 563064, 836160, 2211627, 3283731, 4875451, 12892260, 19140940, 28418164, 75143551, 111563527, 165635151, 437970664, 650241840, 965394360, 2552682051
OFFSET
1,2
COMMENTS
Also values x of Pythagorean triples (x, x+809, y).
Corresponding values y of solutions (x, y) are in A160203.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (873+232*sqrt(2))/809 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (989043+524338*sqrt(2))/809^2 for n mod 3 = 0.
FORMULA
a(n) = 6*a(n-3)-a(n-6)+1618 for n > 6; a(1)=0, a(2)=264, a(3)=1491, a(4)=2427, a(5)=3811, a(6)=10764.
G.f.: x*(264+1227*x+936*x^2-200*x^3-409*x^4-200*x^5) / ((1-x)*(1-6*x^3 +x^6)).
a(3*k+1) = 809*A001652(k) for k >= 0.
MATHEMATICA
LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 264, 1491, 2427, 3811, 10764, 16180}, 50] (* G. C. Greubel, May 04 2018 *)
PROG
(PARI) {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1618*n+654481), print1(n, ", ")))}
(PARI) x='x+O('x^30); concat([0], Vec(x*(264+1227*x+936*x^2-200*x^3 -409*x^4 -200*x^5)/((1-x)*(1-6*x^3 +x^6)))) \\ G. C. Greubel, May 04 2018
(Magma) I:=[0, 264, 1491, 2427, 3811, 10764, 16180]; [n le 7 select I[n] else Self(n-1) +6*Self(n-3) -6*Self(n-4) -Self(n-6) +Self(n-7): n in [1..30]]; // G. C. Greubel, May 04 2018
CROSSREFS
Cf. A160203, A001652, A115135, A156035 (decimal expansion of 3+2*sqrt(2)), A160204 (decimal expansion of (873+232*sqrt(2))/809), A160205 (decimal expansion of (989043+524338*sqrt(2))/809^2).
Sequence in context: A185764 A253916 A195672 * A014745 A004533 A231301
KEYWORD
nonn,easy
AUTHOR
Mohamed Bouhamida, Jun 03 2007
EXTENSIONS
Edited and two terms added by Klaus Brockhaus, May 18 2009
STATUS
approved