login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A123480 Coefficients of the series giving the best rational approximations to sqrt(3). 9
4, 60, 840, 11704, 163020, 2270580, 31625104, 440480880, 6135107220, 85451020204, 1190179175640, 16577057438760, 230888624967004, 3215863692099300, 44791203064423200, 623860979209825504, 8689262505873133860, 121025814103014048540, 185672134936323545704 (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

Table of n, a(n) for n=1..19.

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

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified September 25 18:20 EDT 2017. Contains 292499 sequences.