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

1,1

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).

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

Mathematica

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

Prog

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

Crossrefs

Cf. A123479, A123480, A029549, A123482.

Sequence in context: A295590 A233220 A377749 * A275570 A307618 A225786

Adjacent sequences: A123475 A123476 A123477 * A123479 A123480 A123481

Keyword

nonn,easy

Author

Gene Ward Smith, Sep 28 2006

Extensions

More terms from Colin Barker, Jun 23 2014

Status

approved