A130647 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+839)^2 = y^2.

0, 60, 2241, 2517, 2821, 15180, 16780, 18544, 90517, 99841, 110121, 529600, 583944, 643860, 3088761, 3405501, 3754717, 18004644, 19850740, 21886120, 104940781, 115700617, 127563681, 611641720, 674354640, 743497644, 3564911217

Offset

1,2

Comments

Also values x of Pythagorean triples (x, x+839, y).

Corresponding values y of solutions (x, y) are in A159896.

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) = (843+58*sqrt(2))/839 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (1760979+1141390*sqrt(2))/839^2 for n mod 3 = 0.

Links

G. C. Greubel, Table of n, a(n) for n = 1..3895

Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Formula

a(n) = 6*a(n-3) -a(n-6) +1678 for n > 6; a(1)=0, a(2)=60, a(3)=2241, a(4)=2517, a(5)=2821, a(6)=15180.

G.f.: x*(60+2181*x+276*x^2-56*x^3-727*x^4-56*x^5)/((1-x)*(1-6*x^3+x^6)).

a(3*k+1) = 839*A001652(k) for k >= 0.

Mathematica

LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 60, 2241, 2517, 2821, 15180, 16780}, 30] (* Harvey P. Dale, Jun 19 2014 *)

Prog

(PARI) {forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+1678*n+703921), print1(n, ", ")))}
(Magma) I:=[0, 60, 2241, 2517, 2821, 15180, 16780]; [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 17 2018

Crossrefs

Cf. A159896, A028871, A118337, A130645, A130646, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159897 (decimal expansion of (843+58*sqrt(2))/839), A159898 (decimal expansion of (1760979+1141390*sqrt(2))/839^2).

Sequence in context: A371600 A054623 A075908 * A062263 A075917 A058929

Adjacent sequences: A130644 A130645 A130646 * A130648 A130649 A130650

Keyword

nonn,easy

Author

Mohamed Bouhamida, Jun 20 2007

Extensions

Edited and two terms added by Klaus Brockhaus, Apr 30 2009

Status

approved