A123478 Coefficients of series giving the best rational approximations to sqrt(7).

48, 12240, 3108960, 789663648, 200571457680, 50944360587120, 12939667017670848, 3286624478127808320, 834789677777445642480, 212033291530993065381648, 53855621259194461161296160, 13679115766543862141903843040, 3474441549080881789582414836048

Offset: 1

Author

Gene Ward Smith, Sep 28 2006

Keywords

• nonn
• easy

Comments

The partial sums of the series 8/3 - 1/a(1) - 1/a(2) - 1/a(3) - ... give the best rational approximations to sqrt(7), which constitute every fourth convergent of the continued fraction. The corresponding continued fractions are [2;1,1,1], [2;1,1,1,4,1,1,1], [2;1,1,1,4,1,1,1,4,1,1,1] and so forth.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..416
• Index entries for linear recurrences with constant coefficients, signature (255,-255,1).

Crossrefs

Cf. A123479, A123480, A029549, A123482.

Programs

Mathematica

LinearRecurrence[{255,-255,1},{48,12240,3108960},30] (* Harvey P. Dale, Nov 20 2016 *)

PARI

Vec(-48*x/((x-1)*(x^2-254*x+1)) + O(x^100)) \\ Colin Barker, Jun 23 2014

Formula

a(n+3) = 255 a(n+2) - 255 a(n+1) + a(n).

a(n) = -4/21 + (2/21+1/28*7^(1/2))*(127+48*7^(1/2))^n + (2/21-1/28*7^(1/2))*(127-48*7^(1/2))^n.

G.f.: -48*x / ((x-1)*(x^2-254*x+1)). - Colin Barker, Jun 23 2014

Extensions

More terms from Colin Barker, Jun 23 2014