A218735 - OEIS

%I #57 Nov 30 2024 17:37:55

%S 5,6,9,13,22,33,57,86,149,225,390,589,1021,1542,2673,4037,6998,10569,

%T 18321,27670,47965,72441,125574,189653,328757,496518,860697,1299901,

%U 2253334,3403185,5899305,8909654,15444581,23325777,40434438,61067677,105858733

%N Values of x in the solutions to x^2 - 3xy + y^2 + 29 = 0, where 0 < x < y.

%C The corresponding values of y are given by a(n+2).

%C Positive values of x (or y) satisfying x^2 - 18xy + y^2 + 1856 = 0.

%H Colin Barker, <a href="/A218735/b218735.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-1).

%F a(n) = 3*a(n-2)-a(n-4).

%F G.f.: -x*(x-1)*(5*x^2+11*x+5) / ((x^2-x-1)*(x^2+x-1)).

%e 13 is in the sequence because (x, y) = (13, 33) is a solution to x^2 - 3xy + y^2 + 29 = 0.

%t LinearRecurrence[{0,3,0,-1},{5,6,9,13},40] (* _Harvey P. Dale_, Nov 30 2024 *)

%o (PARI) Vec(-x*(x-1)*(5*x^2+11*x+5)/((x^2-x-1)*(x^2+x-1)) + O(x^100))

%Y Cf. A001519, A005248, A055819, A237132, A237133.

%K nonn,easy

%O 1,1

%A _Colin Barker_, Feb 05 2014