A123480 Coefficients of the series giving the best rational approximations to sqrt(3).

4, 60, 840, 11704, 163020, 2270580, 31625104, 440480880, 6135107220, 85451020204, 1190179175640, 16577057438760, 230888624967004, 3215863692099300, 44791203064423200, 623860979209825504, 8689262505873133860, 121025814103014048540, 1685672134936323545704

Offset: 1

Author

Gene Ward Smith, Sep 28 2006

Keywords

• nonn
• easy

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).

Crossrefs

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

Programs

Mathematica

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

PARI

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

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

a(n) = A001353(n)*A001353(n+1). - Antonio Alberto Olivares, Apr 06 2020