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A237254 Values of x in the solutions to x^2 - 5xy + y^2 + 5 = 0, where 0 < x < y.

%I A237254 #31 Aug 24 2024 15:58:37
%S A237254 1,2,3,9,14,43,67,206,321,987,1538,4729,7369,22658,35307,108561,
%T A237254 169166,520147,810523,2492174,3883449,11940723,18606722,57211441,
%U A237254 89150161,274116482,427144083,1313370969,2046570254,6292738363,9805707187,30150320846,46981965681
%N A237254 Values of x in the solutions to x^2 - 5xy + y^2 + 5 = 0, where 0 < x < y.
%C A237254 The corresponding values of y are given by a(n+2).
%C A237254 Also the solutions to 21x^2-20 is a perfect square. - _Jaimal Ichharam_, Jul 13 2014
%H A237254 Vincenzo Librandi, <a href="/A237254/b237254.txt">Table of n, a(n) for n = 1..1000</a>
%H A237254 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,5,0,-1).
%F A237254 a(n) = 5*a(n-2)-a(n-4).
%F A237254 G.f.: -x*(x-1)*(x^2+3*x+1) / (x^4-5*x^2+1).
%e A237254 9 is in the sequence because (x, y) = (9, 43) is a solution to x^2 - 5xy + y^2 + 5 = 0.
%p A237254 A237254 := proc(n)
%p A237254     coeftayl( -x*(x-1)*(x^2+3*x+1) / (x^4-5*x^2+1), x=0, n);
%p A237254 end proc:
%p A237254 seq(A237254(n), n=1..40); # _Wesley Ivan Hurt_, Jul 14 2014
%t A237254 Rest[CoefficientList[Series[- x (x - 1) (x^2 + 3 x + 1)/(x^4 - 5 x^2 + 1), {x, 0, 40}], x]] (* _Vincenzo Librandi_, Jul 01 2014 *)
%t A237254 LinearRecurrence[{0,5,0,-1},{1,2,3,9},40] (* _Harvey P. Dale_, Aug 24 2024 *)
%o A237254 (PARI) Vec(-x*(x-1)*(x^2+3*x+1)/(x^4-5*x^2+1) + O(x^100))
%Y A237254 Cf. A004253, A237255.
%K A237254 nonn,easy
%O A237254 1,2
%A A237254 _Colin Barker_, Feb 05 2014