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Numerators of continued fraction convergents to sqrt(689).
2

%I #16 Sep 08 2022 08:44:55

%S 26,105,5486,22049,1152034,4630185,241921654,972316801,50802395306,

%T 204181898025,10668261092606,42877226268449,2240284027051954,

%U 9004013334476265,470448977419817734,1890799923013747201

%N Numerators of continued fraction convergents to sqrt(689).

%H Vincenzo Librandi, <a href="/A042324/b042324.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0, 210, 0, -1).

%F G.f.: (26 +105*x +26*x^2 -x^3)/(1 -210*x^2 +x^4). - _Vincenzo Librandi_ Nov 21 2013

%F a(n) = 210*a(n-2) - a(n-4). - _Vincenzo Librandi_, Nov 21 2013

%t Numerator[Convergents[Sqrt[689], 30]] (* or *) CoefficientList[Series[(26 + 105 x + 26 x^2 - x^3)/(1 -210 x^2 + x^4), {x, 0, 30}], x] (* _Vincenzo Librandi_, Nov 21 2013 *)

%t LinearRecurrence[{0,210,0,-1},{26,105,5486,22049},20] (* _Harvey P. Dale_, May 12 2017 *)

%o (Magma) I:=[26, 105, 5486, 22049]; [n le 4 select I[n] else 210*Self(n-2)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Nov 21 2013

%Y Cf. A042325.

%K nonn,cofr,frac,easy

%O 0,1

%A _N. J. A. Sloane_.