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A237132 Values of x in the solutions to x^2 - 3xy + y^2 + 11 = 0, where 0 < x < y.

%I A237132 #26 Sep 03 2019 09:54:46
%S A237132 3,4,5,9,12,23,31,60,81,157,212,411,555,1076,1453,2817,3804,7375,9959,
%T A237132 19308,26073,50549,68260,132339,178707,346468,467861,907065,1224876,
%U A237132 2374727,3206767,6217116,8395425,16276621,21979508,42612747,57543099,111561620
%N A237132 Values of x in the solutions to x^2 - 3xy + y^2 + 11 = 0, where 0 < x < y.
%C A237132 The corresponding values of y are given by a(n+2).
%C A237132 Positive values of x (or y) satisfying x^2 - 18xy + y^2 + 704 = 0.
%H A237132 Colin Barker, <a href="/A237132/b237132.txt">Table of n, a(n) for n = 1..1000</a>
%H A237132 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-1).
%F A237132 a(n) = 3*a(n-2)-a(n-4).
%F A237132 G.f.: -x*(x-1)*(3*x^2+7*x+3) / ((x^2-x-1)*(x^2+x-1)).
%F A237132 a(n) = F(n+2) + (-1)^n*F(n-3), n>1, with F the Fibonacci numbers (A000045). - _Ralf Stephan_, Feb 05 2014
%F A237132 Let h(n) = hypergeom([(1 - n)/2, n mod 2 - n/2], [1 - n], -4) then a(n) = h(n-1) + h(n) for n > 3. - _Peter Luschny_, Sep 03 2019
%e A237132 9 is in the sequence because (x, y) = (9, 23) is a solution to x^2 - 3xy + y^2 + 11 = 0.
%o A237132 (PARI) Vec(-x*(x-1)*(3*x^2+7*x+3)/((x^2-x-1)*(x^2+x-1)) + O(x^100))
%Y A237132 Cf. A001519, A005248, A055819, A237133, A218735.
%K A237132 nonn,easy
%O A237132 1,1
%A A237132 _Colin Barker_, Feb 04 2014