A041030 Numerators of continued fraction convergents to sqrt(20).

4, 9, 76, 161, 1364, 2889, 24476, 51841, 439204, 930249, 7881196, 16692641, 141422324, 299537289, 2537720636, 5374978561, 45537549124, 96450076809, 817138163596, 1730726404001, 14662949395604

Offset

0,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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

Formula

a(2n-1) = ceiling(1/(4/(Fibonacci(6n)*sqrt(5)-Lucas(6n)+2)-2)), a(2n) = ceiling(1/(1-2/(Fibonacci(6n+3)*sqrt(5)-Lucas(6n+3)+2))-2). - Thomas Baruchel

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

From Gerry Martens, Jul 11 2015: (Start)

Interspersion of 2 sequences [a0(n),a1(n)] for n>0 :

a0(n) = -((2+sqrt(5))/(9+4*sqrt(5))^n)+(-2+sqrt(5))*(9+4*sqrt(5))^n.

a1(n) = (1/(9+4*sqrt(5))^n+(9+4*sqrt(5))^n)/2. (End)

Mathematica

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[20], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011 *)
Numerator[Convergents[Sqrt[20], 30]] (* Vincenzo Librandi, Oct 28 2013 *)
a0[n_] := -((2+Sqrt[5])/(9+4*Sqrt[5])^n)+(-2+Sqrt[5])*(9+4*Sqrt[5])^n //Simplify
a1[n_] := (1/(9+4*Sqrt[5])^n+(9+4*Sqrt[5])^n)/2 // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &, Range[20]]] (* Gerry Martens, Jul 11 2015 *)

Crossrefs

Cf. A010476, A041031, A087953.

Sequence in context: A100517 A327579 A041597 * A361067 A359654 A061104

Adjacent sequences: A041027 A041028 A041029 * A041031 A041032 A041033

Keyword

nonn,cofr,frac,easy

Author

N. J. A. Sloane

Status

approved