A123478 - OEIS

%I #18 Mar 17 2024 05:53:28

%S 48,12240,3108960,789663648,200571457680,50944360587120,

%T 12939667017670848,3286624478127808320,834789677777445642480,

%U 212033291530993065381648,53855621259194461161296160,13679115766543862141903843040,3474441549080881789582414836048

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

%C 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.

%H Harvey P. Dale, <a href="/A123478/b123478.txt">Table of n, a(n) for n = 1..416</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (255,-255,1).

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

%F 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.

%F G.f.: -48*x / ((x-1)*(x^2-254*x+1)). - _Colin Barker_, Jun 23 2014

%t LinearRecurrence[{255,-255,1},{48,12240,3108960},30] (* _Harvey P. Dale_, Nov 20 2016 *)

%o (PARI) Vec(-48*x/((x-1)*(x^2-254*x+1)) + O(x^100)) \\ _Colin Barker_, Jun 23 2014

%Y Cf. A123479, A123480, A029549, A123482.

%K nonn,easy

%O 1,1

%A _Gene Ward Smith_, Sep 28 2006

%E More terms from _Colin Barker_, Jun 23 2014