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A157120 Positive numbers y such that y^2 is of the form x^2+(x+103)^2 with integer x.

%I A157120 #16 Apr 18 2024 10:59:44
%S A157120 73,103,205,233,515,1157,1325,2987,6737,7717,17407,39265,44977,101455,
%T A157120 228853,262145,591323,1333853,1527893,3446483,7774265,8905213,
%U A157120 20087575,45311737,51903385,117078967,264096157,302515097,682386227,1539265205
%N A157120 Positive numbers y such that y^2 is of the form x^2+(x+103)^2 with integer x.
%C A157120 (-48, a(1)) and (A157119(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+103)^2 = y^2.
%H A157120 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-1).
%F A157120 a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1) = 73, a(2) = 103, a(3) = 205, a(4) = 233, a(5) = 515, a(6) = 1157.
%F A157120 G.f.: x*(1-x)*(73+176*x+381*x^2+176*x^3+73*x^4) / (1-6*x^3+x^6).
%F A157120 a(3*k-1) = 103*A001653(k) for k >= 1.
%F A157120 Limit_{n -> oo} a(n)/a(n-3) = 3+2*sqrt(2).
%F A157120 Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2))*(11-3*sqrt(2))^2/(11+3*sqrt(2))^2 for n mod 3 = 1.
%F A157120 Limit_{n -> oo} a(n)/a(n-1) = (11+3*sqrt(2))/(11-3*sqrt(2)) for n mod 3 = {0, 2}.
%e A157120 (-48, a(1)) = (-48, 73) is a solution: (-48)^2+(-48+103)^2 = 2304+3025 = 5329 = 73^2.
%e A157120 (A157119(1), a(2)) = (0, 103) is a solution: 0^2+(0+103)^2 = 10609 = 103^2,
%e A157120 (A157119(3), a(4)) = (105, 233) is a solution: 105^2+(105+103)^2 = 11025+43264 = 54289 = 233^2.
%t A157120 Select[Table[Sqrt[x^2+(x+103)^2],{x,-50,3*10^6}],IntegerQ] (* THe program generates the first 20 terms of the sequence. *) (* or *) LinearRecurrence[ {0,0,6,0,0,-1},{73,103,205,233,515,1157},50](* _Harvey P. Dale_, Aug 19 2020 *)
%o A157120 (PARI) {forstep(n=-48, 1100000000, [1, 3], if(issquare(2*n^2+206*n+10609, &k), print1(k, ",")))}
%Y A157120 Cf. A157119, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157121 (decimal expansion of 11+3*sqrt(2)), A157122 (decimal expansion of 11-3*sqrt(2)), A157123 (decimal expansion of (11+3*sqrt(2))/(11-3*sqrt(2))).
%K A157120 nonn,easy
%O A157120 1,1
%A A157120 _Klaus Brockhaus_, Feb 25 2009
%E A157120 Typo corrected by _Klaus Brockhaus_, Mar 01 2009