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A157297 Positive numbers y such that y^2 is of the form x^2+(x+233)^2 with integer x.

%I A157297 #10 Sep 08 2022 08:45:41
%S A157297 185,233,317,793,1165,1717,4573,6757,9985,26645,39377,58193,155297,
%T A157297 229505,339173,905137,1337653,1976845,5275525,7796413,11521897,
%U A157297 30748013,45440825,67154537,179212553,264848537,391405325,1044527305,1543650397
%N A157297 Positive numbers y such that y^2 is of the form x^2+(x+233)^2 with integer x.
%C A157297 (-57, a(1)) and (A129625(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+233)^2 = y^2.
%C A157297 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A157297 lim_{n -> infinity} a(n)/a(n-1) = (251+66*sqrt(2))/233 for n mod 3 = {0, 2}.
%C A157297 lim_{n -> infinity} a(n)/a(n-1) = (82611+44030*sqrt(2))/233^2 for n mod 3 = 1.
%H A157297 G. C. Greubel, <a href="/A157297/b157297.txt">Table of n, a(n) for n = 1..1000</a>
%H A157297 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-1).
%F A157297 a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=185, a(2)=233, a(3)=317, a(4)=793, a(5)=1165, a(6)=1717.
%F A157297 G.f.: (1-x)*(185 +418*x +735*x^2 +418*x^3 +185*x^4)/(1-6*x^3+x^6).
%F A157297 a(3*k-1) = 233*A001653(k) for k >= 1.
%e A157297 (-57, a(1)) = (-57, 185) is a solution: (-57)^2+(-57+233)^2 = 3249+30976 = 34225 = 185^2.
%e A157297 (A129625(1), a(2)) = (0, 233) is a solution: 0^2+(0+233)^2 = 54289 = 233^2.
%e A157297 (A129625(3), a(4)) = (432, 793) is a solution: 432^2+(432+233)^2 = 186624+442225 = 628849 = 793^2.
%t A157297 LinearRecurrence[{0,0,6,0,0,-1}, {185,233,317,793,1165,1717}, 50] (* _G. C. Greubel_, Mar 29 2018 *)
%o A157297 (PARI) {forstep(n=-60, 1100000000, [3,1], if(issquare(2*n^2+466*n+54289, &k),print1(k, ",")))};
%o A157297 (Magma) I:=[185,233,317,793,1165,1717]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // _G. C. Greubel_, Mar 29 2018
%Y A157297 Cf. A129625, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157298 (decimal expansion of (251+66*sqrt(2))/233), A157299 (decimal expansion of (82611+44030*sqrt(2))/233^2).
%K A157297 nonn,easy
%O A157297 1,1
%A A157297 _Klaus Brockhaus_, Apr 11 2009