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A159548 Positive numbers y such that y^2 is of the form x^2+(x+199)^2 with integer x.

%I A159548 #13 Aug 09 2025 11:05:39
%S A159548 181,199,221,865,995,1145,5009,5771,6649,29189,33631,38749,170125,
%T A159548 196015,225845,991561,1142459,1316321,5779241,6658739,7672081,
%U A159548 33683885,38809975,44716165,196324069,226201111,260624909,1144260529,1318396691
%N A159548 Positive numbers y such that y^2 is of the form x^2+(x+199)^2 with integer x.
%C A159548 (-19,a(1)) and (A129993(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+199)^2 = y^2.
%H A159548 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-1).
%F A159548 a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=181, a(2)=199, a(3)=221, a(4)=865, a(5)=995, a(6)=1145.
%F A159548 G.f.: x*(1-x)*(181+380*x+601*x^2+380*x^3+181*x^4) / (1-6*x^3+x^6).
%F A159548 a(3*k-1) = 199*A001653(k) for k >= 1.
%F A159548 Limit_{n -> oo} a(n)/a(n-3) = 3+2*sqrt(2).
%F A159548 Limit_{n -> oo} a(n)/a(n-1) = (201+20*sqrt(2))/199 for n mod 3 = {0, 2}.
%F A159548 Limit_{n -> oo} a(n)/a(n-1) = (91443+58282*sqrt(2))/199^2 for n mod 3 = 1.
%e A159548 (-19, a(1)) = (-19, 181) is a solution: (-19)^2+(-19+199)^2 = 361+32400 = 32761 = 181^2.
%e A159548 (A129993(1), a(2)) = (0, 199) is a solution: 0^2+(0+199)^2 = 39601 = 199^2.
%e A159548 (A129993(3), a(4)) = (504, 865) is a solution: 504^2+(504+199)^2 = 254016+494209 = 748225 = 865^2.
%t A159548 LinearRecurrence[{0,0,6,0,0,-1},{181,199,221,865,995,1145},30] (* _Harvey P. Dale_, Aug 09 2025 *)
%o A159548 (PARI) {forstep(n=-20, 50000000, [1, 3], if(issquare(2*n^2+398*n+39601, &k), print1(k, ",")))}
%Y A159548 Cf. A129993, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159549 (decimal expansion of (201+20*sqrt(2))/199), A159550 (decimal expansion of (91443+58282*sqrt(2))/199^2).
%K A159548 nonn,easy
%O A159548 1,1
%A A159548 _Klaus Brockhaus_, Apr 14 2009