cp's OEIS Frontend

A160583 Positive numbers y such that y^2 is of the form x^2+(x+521)^2 with integer x.

%I A160583 #8 Jul 31 2015 22:08:38
%S A160583 445,521,629,2041,2605,3329,11801,15109,19345,68765,88049,112741,
%T A160583 400789,513185,657101,2335969,2991061,3829865,13615025,17433181,
%U A160583 22322089,79354181,101608025,130102669,462510061,592214969,758293925,2695706185,3451681789,4419660881
%N A160583 Positive numbers y such that y^2 is of the form x^2+(x+521)^2 with integer x.
%C A160583 (-84, a(1)) and (A129725(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+521)^2 = y^2.
%C A160583 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A160583 lim_{n -> infinity} a(n)/a(n-1) = (537+92*sqrt(2))/521 for n mod 3 = {0, 2}.
%C A160583 lim_{n -> infinity} a(n)/a(n-1) = (520659+314170*sqrt(2))/521^2 for n mod 3 = 1.
%H A160583 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 6, 0, 0, -1).
%F A160583 a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=445, a(2)=521, a(3)=629, a(4)=2041, a(5)=2605, a(6)=3329.
%F A160583 G.f.: (1-x)*(445+966*x+1595*x^2+966*x^3+445*x^4) / (1-6*x^3+x^6).
%F A160583 a(3*k-1) = 521*A001653(k) for k >= 1.
%e A160583 (-84, a(1)) = (-84, 445) is a solution: (-84)^2+(-84+521)^2 = 7056+190969 = 198025 = 445^2.
%e A160583 (A129725(1), a(2)) = (0, 521) is a solution: 0^2+(0+521)^2 = 271441 = 521^2.
%e A160583 (A129725(3), a(4)) = (1159, 2041) is a solution: 1159^2+(1159+521)^2 = 1343281+2822400 = 4165681 = 2041^2.
%t A160583 LinearRecurrence[{0, 0, 6, 0, 0, -1}, {445, 521, 629, 2041, 2605, 3329}, 50] (* _Vladimir Joseph Stephan Orlovsky_, Feb 15 2012 *)
%o A160583 (PARI) {forstep(n=-84, 10000000, [3, 1], if(issquare(2*n^2+1042*n+271441, &k), print1(k, ",")))}
%Y A160583 Cf. A129725, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160584 (decimal expansion of (537+92*sqrt(2))/521), A160585 (decimal expansion of (520659+314170*sqrt(2))/521^2).
%K A160583 nonn
%O A160583 1,1
%A A160583 _Klaus Brockhaus_, Jun 08 2009