A159589 - OEIS

%I #11 Sep 08 2022 08:45:44

%S 421,449,481,2045,2245,2465,11849,13021,14309,69049,75881,83389,

%T 402445,442265,486025,2345621,2577709,2832761,13671281,15023989,

%U 16510541,79682065,87566225,96230485,464421109,510373361,560872369,2706844589

%N Positive numbers y such that y^2 is of the form x^2+(x+449)^2 with integer x.

%C (-29,a(1)) and (A130004(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+449)^2 = y^2.

%C lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).

%C lim_{n -> infinity} a(n)/a(n-1) = (451+30*sqrt(2))/449 for n mod 3 = {0, 2}.

%C lim_{n -> infinity} a(n)/a(n-1) = (507363+329222*sqrt(2))/449^2 for n mod 3 = 1.

%H G. C. Greubel, <a href="/A159589/b159589.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-1).

%F a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=421, a(2)=449, a(3)=481, a(4)=2045, a(5)=2245, a(6)=2465.

%F G.f.: (1-x)*(421+870*x+1351*x^2+870*x^3+421*x^4) / (1-6*x^3+x^6).

%F a(3*k-1) = 449*A001653(k) for k >= 1.

%e (-29, a(1)) = (-29, 421) is a solution: (-29)^2+(-29+449)^2 = 841+176400 = 177241 = 421^2.

%e (A130004(1), a(2)) = (0, 449) is a solution: 0^2+(0+449)^2 = 201601 = 449^2.

%e (A130004(3), a(4)) = (1204, 2045) is a solution: 1204^2+(1204+449)^2 = 1449616+2732409 = 4182025 = 2045^2.

%t LinearRecurrence[{0,0,6,0,0,-1}, {421,449,481,2045,2245,2465}, 50] (* _G. C. Greubel_, May 08 2018 *)

%o (PARI) {forstep(n=-32, 50000000, [3, 1], if(issquare(2*n^2+898*n+201601, &k), print1(k, ",")))}

%o (PARI) x='x+O('x^30); Vec((1-x)*(421+870*x+1351*x^2+870*x^3+421*x^4)/(1- 6*x^3+x^6)) \\ _G. C. Greubel_, May 08 2018

%o (Magma) I:=[421,449,481,2045,2245,2465]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // _G. C. Greubel_, May 08 2018

%Y Cf. A130004, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159590 (decimal expansion of (451+30*sqrt(2))/449), A159591 (decimal expansion of (507363+329222*sqrt(2))/449^2).

%K nonn,easy

%O 1,1

%A _Klaus Brockhaus_, Apr 18 2009