A041010 Numerators of continued fraction convergents to sqrt(8).

2, 3, 14, 17, 82, 99, 478, 577, 2786, 3363, 16238, 19601, 94642, 114243, 551614, 665857, 3215042, 3880899, 18738638, 22619537, 109216786, 131836323, 636562078, 768398401, 3710155682, 4478554083, 21624372014, 26102926097, 126036076402, 152139002499

Offset

0,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..199 [1 removed by Georg Fischer, Jul 01 2019]

Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Formula

a(n) = 6*a(n-2) - a(n-4).

a(2n) = a(2n-1) + a(2n-2), a(2n+1) = 4*a(2n) + a(2n-1).

a(2n) = A001333(2n), a(2n+1) = 2*A001333(2n+1).

G.f.: (2+3*x+2*x^2-x^3)/(1-6*x^2+x^4).

a(n) = A001333(n+1)*A000034(n+1). - R. J. Mathar, Jul 08 2009

From Gerry Martens, Jul 11 2015: (Start)

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

a0(n) = -((3-2*sqrt(2))^n*(1+sqrt(2))) + (-1+sqrt(2))*(3+2*sqrt(2))^n.

a1(n) = ((3-2*sqrt(2))^n + (3+2*sqrt(2))^n)/2. (End)

Mathematica

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[8], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011*)
CoefficientList[Series[(2 + 3*x + 2*x^2 - x^3)/(1 - 6*x^2 + x^4), {x, 0, 30}], x]  (* Vincenzo Librandi, Oct 28 2013 *)
a0[n_] := -((3-2*Sqrt[2])^n*(1+Sqrt[2]))+(-1+Sqrt[2])*(3+2*Sqrt[2])^n // Simplify
a1[n_] := ((3-2*Sqrt[2])^n+(3+2*Sqrt[2])^n)/2 // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &, Range[20]]] (* Gerry Martens, Jul 11 2015 *)

Prog

From M. F. Hasler, Nov 01 2019: (Start)
(PARI) A041010=contfracpnqn(c=contfrac(sqrt(8)), #c)[1, ][^-1] \\ Discard possibly incorrect last element. NB: a(n)=A041010[n+1]! For more terms use:
A041010(n)={n<#A041010|| A041010=extend(A041010, [-1, 0, 6, 0]~, n\.8); A041010[n+1]}
extend(A, c, N)={for(n=#A+1, #A=Vec(A, N), A[n]=A[n-#c..n-1]*c); A} \\ (End)

Crossrefs

Cf. A040005 (continued fraction), A041011 (denominators), A010466 (decimals).

Analog for other sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041008 (m=7), A005667 (m=10), A041014 (m=11), A041016 (m=12), ..., A042934 (m=999), A042936 (m=1000).

Sequence in context: A042367 A100341 A041869 * A041733 A212112 A107083

Adjacent sequences: A041007 A041008 A041009 * A041011 A041012 A041013

Keyword

nonn,cofr,frac,easy

Author

N. J. A. Sloane

Extensions

Entry improved by Michael Somos

Initial term 1 removed and b-file, program and formulas adapted by Georg Fischer, Jul 01 2019

Cross-references added by M. F. Hasler, Nov 02 2019

Status

approved