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A160098 Positive numbers y such that y^2 is of the form x^2+(x+601)^2 with integer x.

%I A160098 #8 Mar 03 2024 15:41:22
%S A160098 425,601,1261,1289,3005,7141,7309,17429,41585,42565,101569,242369,
%T A160098 248081,591985,1412629,1445921,3450341,8233405,8427445,20110061,
%U A160098 47987801,49118749,117210025,279693401,286285049,683150089,1630172605
%N A160098 Positive numbers y such that y^2 is of the form x^2+(x+601)^2 with integer x.
%C A160098 (-297, a(1)) and (A111258(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+601)^2 = y^2.
%C A160098 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A160098 lim_{n -> infinity} a(n)/a(n-1) = (843+418*sqrt(2))/601 for n mod 3 = {0, 2}.
%C A160098 lim_{n -> infinity} a(n)/a(n-1) = (361299+5950*sqrt(2))/601^2 for n mod 3 = 1.
%H A160098 G. C. Greubel, <a href="/A160098/b160098.txt">Table of n, a(n) for n = 1..3875</a>
%H A160098 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 6, 0, 0, -1).
%F A160098 a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=425, a(2)=601, a(3)=1261, a(4)=1289, a(5)=3005, a(6)=7141.
%F A160098 G.f.: (1-x)*(425 +1026*x +2287*x^2 +1026*x^3 +425*x^4)/(1-6*x^3+x^6).
%F A160098 a(3*k-1) = 601*A001653(k) for k >= 1.
%e A160098 (-297, a(1)) = (-297, 425) is a solution: (-297)^2+(-297+601)^2 = 88209+92416 = 180625 = 425^2.
%e A160098 (A111258(1), a(2)) = (0, 601) is a solution: 0^2+(0+601)^2 = 361201 = 601^2.
%e A160098 (A111258(3), a(4)) = (560, 1289) is a solution: 560^2+(560+601)^2 = 313600+1347921 = 1661521 = 1289^2.
%t A160098 LinearRecurrence[{0,0,6,0,0,-1}, {425,601,1261,1289,3005,7141}, 50] (* _G. C. Greubel_, Apr 22 2018 *)
%o A160098 (PARI) {forstep(n=-300, 10000000, [3, 1], if(issquare(2*n^2+1202*n+361201, &k), print1(k, ",")))}
%o A160098 (PARI) x='x+O('x^30); Vec((1-x)*(425 +1026*x +2287*x^2 +1026*x^3 +425*x^4 )/(1-6*x^3+x^6)) \\ _G. C. Greubel_, Apr 22 2018
%o A160098 (Magma) I:=[425,601,1261,1289,3005,7141]; [n le 6 select I[n] else 5*Self(n-3) - Self(n-6): n in [1..30]]; // _G. C. Greubel_, Apr 22 2018
%Y A160098 Cf. A111258, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160099 (decimal expansion of (843+418*sqrt(2))/601), A160100 (decimal expansion of (361299+5950*sqrt(2))/601^2).
%K A160098 nonn
%O A160098 1,1
%A A160098 _Klaus Brockhaus_, May 18 2009