A159809 - OEIS

%I #15 Apr 19 2023 02:32:27

%S 197,223,257,925,1115,1345,5353,6467,7813,31193,37687,45533,181805,

%T 219655,265385,1059637,1280243,1546777,6176017,7461803,9015277,

%U 35996465,43490575,52544885,209802773,253481647,306254033,1222820173,1477399307

%N Positive numbers y such that y^2 is of the form x^2+(x+223)^2 with integer x.

%C (-28, a(1)) and (A130609(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+223)^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) = (227+30*sqrt(2))/223 for n mod 3 = {0, 2}.

%C Lim_{n -> infinity} a(n)/a(n-1) = (105507+65798*sqrt(2))/223^2 for n mod 3 = 1.

%C For the generic case x^2 + (x+p)^2 = y^2 with p = m^2 - 2 a prime number in A028871, m >= 5, the x values are given by the sequence defined by a(n) = 6*a(n-3) - a(n-6) + 2*p with a(1)=0, a(2) = 2*m + 2, a(3) = 3*m^2 - 10*m + 8, a(4) = 3*p, a(5) = 3*m^2 + 10*m + 8, a(6) = 20*m^2 - 58*m + 42. Y values are given by the sequence defined by b(n) = 6*b(n-3) - b(n-6) with b(1) = p, b(2) = m^2 + 2*m + 2, b(3) = 5*m^2 - 14*m + 10, b(4) = 5*p, b(5) = 5m^2 + 14*m + 10, b(6) = 29*m^2 - 82*m + 58. - _Mohamed Bouhamida_, Sep 09 2009

%H G. C. Greubel, <a href="/A159809/b159809.txt">Table of n, a(n) for n = 1..3900</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)=197, a(2)=223, a(3)=257, a(4)=925, a(5)=1115, a(6)=1345.

%F G.f.: (1-x)*(197+420*x+677*x^2+420*x^3+197*x^4) / (1-6*x^3+x^6).

%F a(3*k-1) = 223*A001653(k) for k >= 1.

%e (-28, a(1)) = (-28, 197) is a solution: (-28)^2 + (-28+223)^2 = 784 + 38025 = 38809 = 197^2.

%e (A130609(1), a(2)) = (0, 223) is a solution: 0^2 + (0+223)^2 = 49729 = 223^2.

%e (A130609(3), a(4)) = (533, 925) is a solution: 533^2 + (533+223)^2 = 284089 + 571536 = 855625 = 925^2.

%t LinearRecurrence[{0,0,6,0,0,-1}, {197,223,257,925,1115,1345}, 50] (* _G. C. Greubel_, May 21 2018 *)

%o (PARI) {forstep(n=-28, 10000000, [1, 3], if(issquare(2*n^2+446*n+49729, &k), print1(k, ",")))};

%o (Magma) I:=[197,223,257,925,1115,1345]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // _G. C. Greubel_, May 21 2018

%Y Cf. A130609, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A130610 (decimal expansion of (227+30*sqrt(2))/223), A130611 (decimal expansion of (105507+65798*sqrt(2))/223^2).

%K nonn,easy

%O 1,1

%A _Klaus Brockhaus_, Apr 30 2009