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A123480 Coefficients of the series giving the best rational approximations to sqrt(3). 10
4, 60, 840, 11704, 163020, 2270580, 31625104, 440480880, 6135107220, 85451020204, 1190179175640, 16577057438760, 230888624967004, 3215863692099300, 44791203064423200, 623860979209825504, 8689262505873133860, 121025814103014048540, 1685672134936323545704 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The partial sums of the series 2 - 1/a(1) - 1/a(2) - 1/a(3) - ... give the best rational approximations to sqrt(3), which constitute every second convergent of the continued fraction. The corresponding continued fractions are [1;1,2,1], [1;1,2,1,2,1], [1;1,2,1,2,1,2,1], [1;1,2,1,2,1,2,1,2,1] and so forth.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

FORMULA

a(n+3) = 15*a(n+2) - 15*a(n+1) + a(n).

a(n) = -1/3 + (1/6 + 1/12*3^(1/2))*(7 + 4*3^(1/2))^n + (1/6 - 1/12*3^(1/2))*(7 - 4*3^(1/2))^n.

a(n) = 4*A076139(n) = 2*A217855(n) = 1/2*A045899(n) = 4/3*A076140(n). - Peter Bala, Dec 31 2012

G.f.: -4*x/((x-1)*(x^2-14*x+1)). - Colin Barker, Jan 20 2013

MATHEMATICA

CoefficientList[Series[-4*x/((x - 1)*(x^2 - 14*x + 1)), {x, 0, 50}], x] (* G. C. Greubel, Oct 13 2017 *)

PROG

(PARI) x='x+O('x^50); Vec(-4*x/((x-1)*(x^2-14*x+1))) \\ G. C. Greubel, Oct 13 2017 *)

CROSSREFS

Cf. A123478, A123479, A029549, A123482. A045899, A076139, A076140, A217855.

Sequence in context: A034866 A055315 A013482 * A227528 A156090 A181418

Adjacent sequences:  A123477 A123478 A123479 * A123481 A123482 A123483

KEYWORD

nonn,easy

AUTHOR

Gene Ward Smith, Sep 28 2006

EXTENSIONS

More terms from Colin Barker, Jan 20 2013

STATUS

approved

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Last modified January 17 19:11 EST 2018. Contains 297829 sequences.