A041142 Numerators of continued fraction convergents to sqrt(80).

8, 9, 152, 161, 2728, 2889, 48952, 51841, 878408, 930249, 15762392, 16692641, 282844648, 299537289, 5075441272, 5374978561, 91075098248, 96450076809, 1634276327192, 1730726404001, 29325898791208, 31056625195209, 526231901914552, 557288527109761

Offset

0,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..199

Index entries for linear recurrences with constant coefficients, signature (0,18,0,-1).

Formula

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

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

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

a(n) = (5 - 3*(-1)^(n+1))*Lucas(3*(n+1))/4. - Ehren Metcalfe, Apr 15 2019

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A010532, A041143.

Sequence in context: A165467 A239584 A041141 * A307947 A175849 A294468

Adjacent sequences: A041139 A041140 A041141 * A041143 A041144 A041145

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Extensions

More terms from Colin Barker, Nov 05 2013

First term 1 removed in b-file, formulas and programs by Georg Fischer, Jul 01 2019

Status

approved