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A161486 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+191)^2 = y^2.

%I A161486 #8 Mar 29 2016 10:38:25
%S A161486 0,69,336,573,936,2449,3820,5929,14740,22729,35020,86373,132936,
%T A161486 204573,503880,775269,1192800,2937289,4519060,6952609,17120236,
%U A161486 26339473,40523236,99784509,153518160,236187189,581587200,894769869,1376600280
%N A161486 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+191)^2 = y^2.
%C A161486 Corresponding values y of solutions (x, y) are in A161487.
%C A161486 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A161486 lim_{n -> infinity} a(n)/a(n-1) = (209+60*sqrt(2))/191 for n mod 3 = {1, 2}.
%C A161486 lim_{n -> infinity} a(n)/a(n-1) = (52323+26522*sqrt(2))/191^2 for n mod 3 = 0.
%H A161486 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,6,-6,0,-1,1).
%F A161486 a(n) = 6*a(n-3)-a(n-6)+382 for n > 6; a(1)=0, a(2)=69, a(3)=336, a(4)=573, a(5)=936, a(6)=2449.
%F A161486 G.f.: x*(69+267*x+237*x^2-51*x^3-89*x^4-51*x^5) / ((1-x)*(1-6*x^3+x^6)).
%F A161486 a(3*k+1) = 191*A001652(k) for k >= 0.
%t A161486 Transpose[NestList[Flatten[{Rest[#],6#[[4]]-First[#]+382}]&,{0,69, 336, 573, 936,2449},40]][[1]]  (* _Harvey P. Dale_, Apr 01 2011 *)
%t A161486 LinearRecurrence[{1,0,6,-6,0,-1,1},{0,69,336,573,936,2449,3820},40] (* _Harvey P. Dale_, Mar 29 2016 *)
%o A161486 (PARI) {forstep(n=0, 10000000, [1, 3], if(issquare(2*n^2+382*n+36481), print1(n, ",")))}
%Y A161486 Cf. A161487, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A161488 (decimal expansion of (209+60*sqrt(2))/191), A161489 (decimal expansion of (52323+26522*sqrt(2))/191^2).
%K A161486 nonn
%O A161486 1,2
%A A161486 _Klaus Brockhaus_, Jun 13 2009