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%I #4 Feb 15 2020 10:52:27
%S 17,23,37,65,115,205,373,667,1193,2173,3887,6953,12665,22655,40525,
%T 73817,132043,236197,430237,769603,1376657,2507605,4485575,8023745,
%U 14615393,26143847,46765813,85184753,152377507,272571133,496493125
%N Positive numbers y such that y^2 is of the form x^2+(x+23)^2 with integer x.
%C (-8, a(1)) and(A118337(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+23)^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) = (27+10*sqrt(2))/23 for n mod 3 = {0, 2}.
%C lim_{n -> infinity} a(n)/a(n-1) = (627+238*sqrt(2))/23^2 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)+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_, Sep 09 2009]
%F a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=17, a(2)=23, a(3)=37, a(4)=65, a(5)=115, a(6)=205.
%F G.f.: x*(1-x)*(17+40*x+77*x^2+40*x^3+17*x^4)/(1-6*x^3+x^6).
%e (-8, a(1)) = (-8, 17) is a solution: (-8)^2+(-8+23)^2 = 64+225 = 289 = 17^2.
%e (A118337(1), a(2)) = (0, 23) is a solution: 0^2+(0+23)^2 = 529 = 23^2.
%e (A118337(3), a(4)) = (33, 65) is a solution: 33^2+(33+23)^2 = 1089+3136 = 4225 = 65^2.
%o (PARI) {forstep(n=-8, 360000000, [1,3], if(issquare(2*n*(n+23)+529, &k), print1(k, ",")))}
%Y Cf. A118337, A156035 (decimal expansion of 3+2*sqrt(2)), A156571 (decimal expansion of (27+10*sqrt(2))/23), A157472 (decimal expansion of (627+238*sqrt(2))/23^2).
%Y A156570 (first trisection), A156568 (second trisection), A156569 (third trisection).
%K nonn
%O 1,1
%A _Klaus Brockhaus_, Feb 11 2009 , Feb 16 2009
%E G.f. corrected, third and fourth comment edited, cross-reference added by _Klaus Brockhaus_, Sep 18 2009
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