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A160090 Positive numbers y such that y^2 is of the form x^2 + (x + 569)^2 with integer x.

%I A160090 #9 Sep 08 2022 08:45:44
%S A160090 485,569,689,2221,2845,3649,12841,16501,21205,74825,96161,123581,
%T A160090 436109,560465,720281,2541829,3266629,4198105,14814865,19039309,
%U A160090 24468349,86347361,110969225,142611989,503269301,646776041,831203585,2933268445
%N A160090 Positive numbers y such that y^2 is of the form x^2 + (x + 569)^2 with integer x.
%C A160090 (-93, a(1)) and (A101152(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+569)^2 = y^2.
%C A160090 Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A160090 Lim_{n -> infinity} a(n)/a(n-1) = (587+102*sqrt(2))/569 for n mod 3 = {0, 2}.
%C A160090 Lim_{n -> infinity} a(n)/a(n-1) = (617139+371510*sqrt(2))/569^2 for n mod 3 = 1.
%H A160090 G. C. Greubel, <a href="/A160090/b160090.txt">Table of n, a(n) for n = 1..1000</a>
%H A160090 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-1).
%F A160090 a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=485, a(2)=569, a(3)=689, a(4)=2221, a(5)=2845, a(6)=3649.
%F A160090 G.f.: (1-x)*(485 +1054*x +1743*x^2 +1054*x^3 +485*x^4) / (1-6*x^3+x^6).
%F A160090 a(3*k-1) = 569*A001653(k) for k >= 1.
%e A160090 (-93, a(1)) = (-93, 485) is a solution: (-93)^2+(-93+569)^2 = 8649+226576 = 235225 = 485^2.
%e A160090 (A101152(1), a(2)) = (0, 569) is a solution: 0^2+(0+569)^2 = 323761= 569^2.
%e A160090 (A101152(3), a(4)) = (1260, 2221) is a solution: 1260^2+(1260+569)^2 = 1587600+3345241 = 4932841 = 2221^2.
%t A160090 LinearRecurrence[{0,0,6,0,0,-1}, {485,569,689,2221,2845,3649}, 50] (* _G. C. Greubel_, Apr 21 2018 *)
%o A160090 (PARI) {forstep(n=-96, 10000000, [3, 1], if(issquare(2*n^2+1138*n+323761, &k), print1(k, ",")))}
%o A160090 (PARI) x='x+O('x^30); Vec((1-x)*(485 +1054*x +1743*x^2 +1054*x^3 +485*x^4)/(1-6*x^3+x^6)) \\ _G. C. Greubel_, Apr 21 2018
%o A160090 (Magma) I:=[485,569,689,2221,2845,3649]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // _G. C. Greubel_, Apr 21 2018
%Y A160090 Cf. A101152, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160091 (decimal expansion of (587+102*sqrt(2))/569), A160092 (decimal expansion of (617139+371510*sqrt(2))/569^2).
%K A160090 nonn
%O A160090 1,1
%A A160090 _Klaus Brockhaus_, May 04 2009