A123654 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+809)^2 = y^2.

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

Author

Mohamed Bouhamida, Jun 03 2007

Keywords

• nonn
• easy

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..2500
• Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

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).

Programs

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

Mathematica

LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,264,1491,2427,3811,10764,16180}, 50] (* G. C. Greubel, May 04 2018 *)

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

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.

Extensions

Edited and two terms added by Klaus Brockhaus, May 18 2009