A122694 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+761)^2 = y^2.

0, 583, 820, 2283, 5440, 6783, 15220, 33579, 41400, 90559, 197556, 243139, 529656, 1153279, 1418956, 3088899, 6723640, 8272119, 18005260, 39190083, 48215280, 104944183, 228418380, 281021083, 611661360, 1331321719, 1637912740

Offset: 1

Author

Mohamed Bouhamida, Jun 03 2007

Keywords

• nonn
• easy

Comments

Also values x of Pythagorean triples (x, x+761, y).

Corresponding values y of solutions (x, y) are in A160200.

lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2) = A156035.

lim_{n -> infinity} a(n)/a(n-1) = (1003+462*sqrt(2))/761 = A160201 for n mod 3 = {1, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (591603+85478*sqrt(2))/761^2 = A160202 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. A160200, A001652, A115135.

Programs

Magma

I:=[0,583,820,2283,5440,6783,15220]; [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, 583, 820, 2283, 5440, 6783, 15220}, 27] (* Jean-François Alcover, Nov 13 2017 *)

PARI

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1522*n+579121), print1(n, ",")))}

PARI

x='x+O('x^30); concat([0], Vec(x*(583 +237*x +1463*x^2 -341*x^3 -79*x^4 -341*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) +1522 for n > 6; a(1)=0, a(2)=583, a(3)=820, a(4)=2283, a(5)=5440, a(6)=6783.

G.f.: x*(583 +237*x +1463*x^2 -341*x^3 -79*x^4 -341*x^5)/((1-x)*(1 -6*x^3 +x^6)).

a(3*k+1) = 761*A001652(k) for k >= 0.

Extensions

Edited and one term added by Klaus Brockhaus, May 18 2009