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%I #9 Jul 14 2015 09:39:05
%S 65,89,149,241,445,829,1381,2581,4825,8045,15041,28121,46889,87665,
%T 163901,273289,510949,955285,1592845,2978029,5567809,9283781,17357225,
%U 32451569,54109841,101165321,189141605,315375265,589634701,1102398061
%N Positive numbers y such that y^2 is of the form x^2+(x+89)^2 with integer x.
%C (-33, a(1)) and (A129298(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+89)^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) = (107+42*sqrt(2))/89 for n mod 3 = {0, 2}.
%C lim_{n -> infinity} a(n)/a(n-1) = (8979+2990*sqrt(2))/89^2 for n mod 3 = 1.
%H Harvey P. Dale, <a href="/A160055/b160055.txt">Table of n, a(n) for n = 1..1000</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)=65, a(2)=89, a(3)=149, a(4)=241, a(5)=445, a(6)=829.
%F G.f.: (1-x)*(65+154*x+303*x^2+154*x^3+65*x^4) / (1-6*x^3+x^6).
%F a(3*k-1) = 89*A001653(k) for k >= 1.
%e (-33, a(1)) = (-33, 65) is a solution: (-33)^2+(-33+89)^2 = 1089+3136 = 4225 = 65^2.
%e (A129298(1), a(2)) = (0, 89) is a solution: 0^2+(0+89)^2 = 7921 = 89^2.
%e (A129298(3), a(4)) = (120, 241) is a solution: 120^2+(120+89)^2 = 14400+43681 = 58081 = 241^2.
%t LinearRecurrence[{0,0,6,0,0,-1},{65,89,149,241,445,829},40] (* _Harvey P. Dale_, Feb 04 2015 *)
%o (PARI) {forstep(n=-36, 10000000, [3, 1], if(issquare(2*n^2+178*n+7921, &k), print1(k, ",")))}
%Y Cf. A129298, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160056 (decimal expansion of (107+42*sqrt(2))/89), A160057 (decimal expansion of (8979+2990*sqrt(2))/89^2).
%K nonn
%O 1,1
%A _Klaus Brockhaus_, May 04 2009
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