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
KEYWORD
nonn,easy
AUTHOR
Gene Ward Smith, Sep 28 2006
EXTENSIONS
More terms from Colin Barker, Jun 23 2014
STATUS
approved