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%I #6 Sep 08 2022 08:45:44
%S 533,617,733,2465,3085,3865,14257,17893,22457,83077,104273,130877,
%T 484205,607745,762805,2822153,3542197,4445953,16448713,20645437,
%U 25912913,95870125,120330425,151031525,558772037,701337113,880276237
%N Positive numbers y such that y^2 is of the form x^2+(x+617)^2 with integer x.
%C (-92, a(1)) and (A115135(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+617)^2 = y^2.
%C lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C lim_{n -> infinity} a(n)/a(n-1) = (633+100*sqrt(2))/617 for n mod 3 = {0, 2}.
%C lim_{n -> infinity} a(n)/a(n-1) = (755667+461578*sqrt(2))/617^2 for n mod 3 = 1.
%H G. C. Greubel, <a href="/A160176/b160176.txt">Table of n, a(n) for n = 1..2500</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-1).
%F a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=533, a(2)=617, a(3)=733, a(4)=2465, a(5)=3085, a(6)=3865.
%F G.f.: (1-x)*(533 +1150*x +1883*x^2 +1150*x^3 +533*x^4)/(1-6*x^3+x^6).
%F a(3*k-1) = 617*A001653(k) for k >= 1.
%e (-92, a(1)) = (-92, 533) is a solution: (-92)^2+(-92+617)^2 = 8464+275625 = 284089 = 533^2.
%e (A115135(1), a(2)) = (0, 617) is a solution: 0^2+(0+617)^2 = 380689 = 617^2.
%e (A115135(3), a(4)) = (1407, 2465) is a solution: 1407^2+(1407+617)^2 = 1979649+4096576 = 6076225 = 2465^2.
%t LinearRecurrence[{0,0,6,0,0,-1}, {533,617,733,2465,3085,3865}, 50] (* _G. C. Greubel_, May 04 2018 *)
%o (PARI) {forstep(n=-92, 10000000, [3, 1], if(issquare(2*n^2+1234*n+380689, &k), print1(k, ",")))}
%o (PARI) x='x+O('x^30); Vec((1-x)*(533 +1150*x +1883*x^2 +1150*x^3 +533*x^4)/(1-6*x^3+x^6)) \\ _G. C. Greubel_, May 04 2018
%o (Magma) I:=[533,617,733,2465,3085,3865]; [n le 6 select I[n] else 6*Self(n31) -Self(n-6): n in [1..30]]; // _G. C. Greubel_, May 04 2018
%Y Cf. A115135, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160177 (decimal expansion of (633+100*sqrt(2))/617), A160178 (decimal expansion of (755667+461578*sqrt(2))/617^2).
%K nonn
%O 1,1
%A _Klaus Brockhaus_, May 18 2009
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