Search: id:A159893
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%I A159893
%S A159893 677,727,785,3277,3635,4033,18985,21083,23413,110633,122863,136445,
%T A159893 644813,716095,795257,3758245,4173707,4635097,21904657,24326147,
%U A159893 27015325,127669697,141783175,157456853,744113525,826372903,917725793
%N A159893 Positive numbers y such that y^2 is of the form x^2+(x+727)^2 with integer 
               x.
%C A159893 (-52, a(1)) and (A130646(n), a(n+1)) are solutions (x, y) to the Diophantine 
               equation x^2+(x+727)^2 = y^2.
%C A159893 lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
%C A159893 lim_{n -> infinity} a(n)/a(n-1) = (731+54*sqrt(2))/727 for n mod 3 = 
               {0, 2}.
%C A159893 lim_{n -> infinity} a(n)/a(n-1) = (1304787+843542*sqrt(2))/727^2 for 
               n mod 3 = 1.
%C A159893 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)+2p 
               with a(1)=0, a(2)=2m+2, a(3)=3m^2-10m+8, a(4)=3p, a(5)=3m^2+10m+8, 
               a(6)=20m^2-58m+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+2m+2, b(3)=5m^2-14m+10, 
               b(4)=5p, b(5)=5m^2+14m+10, b(6)=29m^2-82m+58. [From Mohamed Bouhamida 
               (bhmd95(AT)yahoo.fr), Sep 09 2009]
%F A159893 a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=677, a(2)=727, a(3)=785, a(4)=3277, 
               a(5)=3635, a(6)=4033.
%F A159893 G.f.: (1-x)*(677+1404*x+2189*x^2+1404*x^3+677*x^4) / (1-6*x^3+x^6).
%F A159893 a(3*k-1) = 727*A001653(k) for k >= 1.
%e A159893 (-52, a(1)) = (-52, 677) is a solution: (-52)^2+(-52+727)^2 = 2704+455625 
               = 458329 = 677^2.
%e A159893 (A130646(1), a(2)) = (0, 727) is a solution: 0^2+(0+727)^2 = 528529 = 
               727^2.
%e A159893 (A130646(3), a(4)) = (1925, 3277) is a solution: 1925^2+(1925+727)^2 
               = 3705625+7033104 = 10738729 = 3277^2.
%o A159893 (PARI) {forstep(n=-52, 10000000, [1, 3], if(issquare(2*n^2+1454*n+528529, 
               &k), print1(k, ",")))}
%Y A159893 Cf. A130646, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159894 
               (decimal expansion of (731+54*sqrt(2))/727), A159895 (decimal expansion 
               of (1304787+843542*sqrt(2))/727^2).
%Y A159893 Sequence in context: A014759 A058462 A058450 this_sequence A142755 A158386 
               A031614
%Y A159893 Adjacent sequences: A159890 A159891 A159892 this_sequence A159894 A159895 
               A159896
%K A159893 nonn
%O A159893 1,1
%A A159893 Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Apr 30 2009

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