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A041010 Numerators of continued fraction convergents to sqrt(8). 7
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 26 14:21 EDT 2022. Contains 356999 sequences. (Running on oeis4.)