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A159890 Positive numbers y such that y^2 is of the form x^2+(x+439)^2 with integer x.

%I A159890 #11 Sep 08 2022 08:45:44
%S A159890 401,439,485,1921,2195,2509,11125,12731,14569,64829,74191,84905,
%T A159890 377849,432415,494861,2202265,2520299,2884261,12835741,14689379,
%U A159890 16810705,74812181,85615975,97979969,436037345,499006471,571069109,2541411889
%N A159890 Positive numbers y such that y^2 is of the form x^2+(x+439)^2 with integer x.
%C A159890 (-40, a(1)) and (A130645(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+439)^2 = y^2.
%C A159890 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A159890 lim_{n -> infinity} a(n)/a(n-1) = (443+42*sqrt(2))/439 for n mod 3 = {0, 2}.
%C A159890 lim_{n -> infinity} a(n)/a(n-1) = (450483+287918*sqrt(2))/439^2 for n mod 3 = 1.
%C A159890 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)= 5*m^2 +14*m +10, b(6)= 29*m^2 -82*m +58. - _Mohamed Bouhamida_, Sep 09 2009
%H A159890 G. C. Greubel, <a href="/A159890/b159890.txt">Table of n, a(n) for n = 1..3895</a>
%H A159890 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,6,0,0,-1).
%F A159890 a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=401, a(2)=439, a(3)=485, a(4)=1921, a(5)=2195, a(6)=2509.
%F A159890 G.f.: (1-x)*(401+840*x+1325*x^2+840*x^3+401*x^4) / (1-6*x^3+x^6).
%F A159890 a(3*k-1) = 439*A001653(k) for k >= 1.
%e A159890 (-40, a(1)) = (-40, 401) is a solution: (-40)^2+(-40+439)^2 = 1600+159201 = 160801 = 401^2.
%e A159890 (A130645(1), a(2)) = (0, 439) is a solution: 0^2+(0+439)^2 = 192721 = 439^2.
%e A159890 (A130645(3), a(4)) = (1121, 1921) is a solution: 1121^2+(1121+439)^2 = 1256641+2433600 = 3690241 = 1921^2.
%t A159890 LinearRecurrence[{0,0,6,0,0,-1}, {401,439,485,1921,2195,2509}, 50] (* _G. C. Greubel_, May 17 2018 *)
%o A159890 (PARI) {forstep(n=-40, 10000000, [1, 3], if(issquare(2*n^2+878*n+192721, &k), print1(k, ",")))}
%o A159890 (Magma) I:=[401,439,485,1921,2195,2509]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // _G. C. Greubel_, May 17 2018
%Y A159890 Cf. A130645, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159891 (decimal expansion of (443+42*sqrt(2))/439), A159892 (decimal expansion of (450483+287918*sqrt(2))/439^2).
%K A159890 nonn,easy
%O A159890 1,1
%A A159890 _Klaus Brockhaus_, Apr 30 2009