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

%I A122694 #21 Sep 08 2022 08:45:28
%S A122694 0,583,820,2283,5440,6783,15220,33579,41400,90559,197556,243139,
%T A122694 529656,1153279,1418956,3088899,6723640,8272119,18005260,39190083,
%U A122694 48215280,104944183,228418380,281021083,611661360,1331321719,1637912740
%N A122694 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+761)^2 = y^2.
%C A122694 Also values x of Pythagorean triples (x, x+761, y).
%C A122694 Corresponding values y of solutions (x, y) are in A160200.
%C A122694 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2) = A156035.
%C A122694 lim_{n -> infinity} a(n)/a(n-1) = (1003+462*sqrt(2))/761 = A160201 for n mod 3 = {1, 2}.
%C A122694 lim_{n -> infinity} a(n)/a(n-1) = (591603+85478*sqrt(2))/761^2 = A160202 for n mod 3 = 0.
%H A122694 G. C. Greubel, <a href="/A122694/b122694.txt">Table of n, a(n) for n = 1..2500</a>
%H A122694 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F A122694 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.
%F A122694 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)).
%F A122694 a(3*k+1) = 761*A001652(k) for k >= 0.
%t A122694 LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 583, 820, 2283, 5440, 6783, 15220}, 27] (* _Jean-François Alcover_, Nov 13 2017 *)
%o A122694 (PARI) {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1522*n+579121), print1(n, ",")))}
%o A122694 (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
%o A122694 (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
%Y A122694 Cf. A160200, A001652, A115135.
%K A122694 nonn,easy
%O A122694 1,2
%A A122694 _Mohamed Bouhamida_, Jun 03 2007
%E A122694 Edited and one term added by _Klaus Brockhaus_, May 18 2009