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Numerators of continued fraction convergents to sqrt(80).
2

%I #31 Sep 08 2022 08:44:54

%S 8,9,152,161,2728,2889,48952,51841,878408,930249,15762392,16692641,

%T 282844648,299537289,5075441272,5374978561,91075098248,96450076809,

%U 1634276327192,1730726404001,29325898791208,31056625195209,526231901914552,557288527109761

%N Numerators of continued fraction convergents to sqrt(80).

%H Vincenzo Librandi, <a href="/A041142/b041142.txt">Table of n, a(n) for n = 0..199</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,18,0,-1).

%F G.f.: (8+9*x+8*x^2-x^3)/(1-18*x^2+x^4).

%F a(n) = 18*a(n-2) - a(n-4).

%F a(n) = (-3*(-2-sqrt(5))^(n+1) + 5*(2-sqrt(5))^(n+1) - 3*(-2+sqrt(5))^(n+1) + 5*(2+sqrt(5))^(n+1))/4. - _Colin Barker_, Mar 27 2016

%F a(n) = (5 - 3*(-1)^(n+1))*Lucas(3*(n+1))/4. - _Ehren Metcalfe_, Apr 15 2019

%t CoefficientList[Series[(8+9*x+8*x^2-x^3)/(1-18*x^2+x^4), {x, 0, 30}], x] (* _Vincenzo Librandi_, Oct 29 2013 *)

%o (PARI) Vec((8+9*x+8*x^2-x^3)/(1-18*x^2+x^4) + O(x^30)) \\ _Colin Barker_, Mar 27 2016

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (8+9*x+8*x^2-x^3)/(1-18*x^2+x^4) )); // _G. C. Greubel_, Apr 16 2019

%o (Sage) ((8+9*x+8*x^2-x^3)/(1-18*x^2+x^4)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 16 2019

%Y Cf. A010532, A041143.

%K nonn,frac,easy

%O 0,1

%A _N. J. A. Sloane_

%E More terms from _Colin Barker_, Nov 05 2013

%E First term 1 removed in b-file, formulas and programs by _Georg Fischer_, Jul 01 2019