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A159750 Positive numbers y such that y^2 is of the form x^2+(x+47)^2 with integer x. 4

%I

%S 37,47,65,157,235,353,905,1363,2053,5273,7943,11965,30733,46295,69737,

%T 179125,269827,406457,1044017,1572667,2369005,6084977,9166175,

%U 13807573,35465845,53424383,80476433,206710093,311380123,469051025,1204794713

%N Positive numbers y such that y^2 is of the form x^2+(x+47)^2 with integer x.

%C (-12, a(1)) and (A118675(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+47)^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) = (51+14*sqrt(2))/47 for n mod 3 = {0, 2}.

%C lim_{n -> infinity} a(n)/a(n-1) = (3267+1702*sqrt(2))/47^2 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>=2, 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)= 5*m^2 +14*m +10, b(6)= 29*m^2 -82*m +58. - _Mohamed Bouhamida_, Sep 09 2009

%H G. C. Greubel, <a href="/A159750/b159750.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).

%H J. Gebel, <a href="/A001014/a001014.txt">Integer points on Mordell curves</a> [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

%F a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=37, a(2)=47, a(3)=65, a(4)=157, a(5)=235, a(6)=353.

%F G.f.: (1-x)*(37+84*x+149*x^2+84*x^3+37*x^4) / (1-6*x^3+x^6).

%F a(3*k-1) = 47*A001653(k) for k >= 1.

%e (-12, a(1)) = (-12, 37) is a solution: (-12)^2+(-12+47)^2 = 144+1225 = 1369 = 37^2.

%e (A118675(1), a(2)) = (0, 47) is a solution: 0^2+(0+47)^2 = 2209 = 47^2.

%e (A118675(3), a(4)) = (85, 157) is a solution: 85^2+(85+47)^2 = 7225+17424 = 24649 = 157^2.

%t LinearRecurrence[{0,0,6,0,0,-1}, {37,47,65,157,235,353}, 50] (* _G. C. Greubel_, May 22 2018 *)

%o (PARI) {forstep(n=-12, 100000000, [1, 3], if(issquare(2*n^2+94*n+2209, &k), print1(k, ",")))};

%o (PARI) x='x+O('x^30); Vec((1-x)*(37+84*x+149*x^2+84*x^3+37*x^4)/(1 -6*x^3 +x^6)) \\ _G. C. Greubel_, May 22 2018

%o (MAGMA) I:=[37,47,65,157,235,353]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // _G. C. Greubel_, May 22 2018

%Y Cf. A118675, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159751 (decimal expansion of (51+14*sqrt(2))/47), A159752 (decimal expansion of (3267+1702*sqrt(2))/47^2).

%K nonn,easy

%O 1,1

%A _Klaus Brockhaus_, Apr 30 2009

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Last modified May 12 03:08 EDT 2021. Contains 343810 sequences. (Running on oeis4.)