A041028 - OEIS

%I #32 Apr 16 2022 01:31:18

%S 4,9,13,48,61,170,1421,3012,4433,16311,20744,57799,483136,1024071,

%T 1507207,5545692,7052899,19651490,164264819,348181128,512445947,

%U 1885518969,2397964916,6681448801,55849555324

%N Numerators of continued fraction convergents to sqrt(19).

%H Vincenzo Librandi, <a href="/A041028/b041028.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,340,0,0,0,0,0,-1).

%F a(n) = 340*a(n-6)-a(n-12). G.f.: -(x^11 -4*x^10 +9*x^9 -13*x^8 +48*x^7 -61*x^6 -170*x^5 -61*x^4 -48*x^3 -13*x^2 -9*x -4)/(x^12-340*x^6+1). [_Colin Barker_, Jul 16 2012]

%t Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[19],n]]],{n,1,50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 17 2011 *)

%t CoefficientList[Series[- (x^11 - 4 x^10 + 9 x^9 - 13 x^8 + 48 x^7 - 61 x^6 - 170 x^5 - 61 x^4 - 48 x^3 - 13 x^2 - 9 x - 4)/(x^12 - 340 x^6 + 1), {x, 0, 30}], x] (* _Vincenzo Librandi_, Oct 22 2013 *)

%t LinearRecurrence[{0,0,0,0,0,340,0,0,0,0,0,-1},{4,9,13,48,61,170,1421,3012,4433,16311,20744,57799},30] (* or *) Numerator[ Convergents[ Sqrt[19],30]] (* _Harvey P. Dale_, Jul 04 2021 *)

%Y Cf. A010124, A010475, A041029.

%K nonn,cofr,frac,easy

%O 0,1

%A _N. J. A. Sloane_.