cp's OEIS Frontend

A160574 Positive numbers y such that y^2 is of the form x^2+(x+313)^2 with integer x.

%I A160574 #4 Dec 21 2022 13:08:37
%S A160574 233,313,493,905,1565,2725,5197,9077,15857,30277,52897,92417,176465,
%T A160574 308305,538645,1028513,1796933,3139453,5994613,10473293,18298073,
%U A160574 34939165,61042825,106648985,203640377,355783657,621595837,1186903097
%N A160574 Positive numbers y such that y^2 is of the form x^2+(x+313)^2 with integer x.
%C A160574 (-105, a(1)) and (A129640(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+313)^2 = y^2.
%C A160574 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A160574 lim_{n -> infinity} a(n)/a(n-1) = (363+130*sqrt(2))/313 for n mod 3 = {0, 2}.
%C A160574 lim_{n -> infinity} a(n)/a(n-1) = (119187+47998*sqrt(2))/313^2 for n mod 3 = 1.
%H A160574 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-1).
%F A160574 a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=233, a(2)=313, a(3)=493, a(4)=905, a(5)=1565, a(6)=2725.
%F A160574 G.f.: (1-x)*(233+546*x+1039*x^2+546*x^3+233*x^4) / (1-6*x^3+x^6).
%F A160574 a(3*k-1) = 313*A001653(k) for k >= 1.
%e A160574 (-105, a(1)) = (-105, 233) is a solution: (-105)^2+(-105+313)^2 = 11025+43264 = 54289 = 233^2.
%e A160574 (A129640(1), a(2)) = (0, 313) is a solution: 0^2+(0+313)^2 = 97969 = 313^2.
%e A160574 (A129640(3), a(4)) = (464, 905) is a solution: 464^2+(464+313)^2 = 215296+603729 = 819025 = 905^2.
%t A160574 LinearRecurrence[{0,0,6,0,0,-1},{233,313,493,905,1565,2725},30] (* _Harvey P. Dale_, Dec 21 2022 *)
%o A160574 (PARI) {forstep(n=-108, 10000000, [3, 1], if(issquare(2*n^2+626*n+97969, &k), print1(k, ",")))}
%Y A160574 Cf. A129640, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160575 (decimal expansion of (363+130*sqrt(2))/313), A160576 (decimal expansion of (119187+47998*sqrt(2))/313^2).
%K A160574 nonn
%O A160574 1,1
%A A160574 _Klaus Brockhaus_, Jun 08 2009