A144533 Numerators of continued fraction convergents to sqrt(8/9).
0, 1, 16, 33, 544, 1121, 18480, 38081, 627776, 1293633, 21325904, 43945441, 724452960, 1492851361, 24610074736, 50713000833, 836018088064, 1722749176961, 28400004919440, 58522759015841, 964764149172896, 1988051057361633, 32773581066959024, 67535213191279681
Offset: 0
Examples
0, 1, 16/17, 33/35, 544/577, 1121/1189, 18480/19601, 38081/40391, 627776/665857, ...
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Index entries for linear recurrences with constant coefficients, signature (0,34,0,-1).
Programs
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Magma
I:=[0, 1, 16, 33]; [n le 4 select I[n] else 34*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 10 2013
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Mathematica
CoefficientList[Series[- x (x^2 - 16 x - 1)/((x^2 - 6 x + 1) (x^2 + 6 x + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 10 2013 *) Convergents[Sqrt[8/9],30]//Numerator (* or *) LinearRecurrence[{0,34,0,-1},{0,1,16,33},30] (* Harvey P. Dale, Oct 09 2022 *)
Formula
a(n) = 16*a(n-1) + a(n-2) if n odd, otherwise a(n) = 2*a(n-1) + a(n-2), for n >= 2.
a(n) = 34*a(n-2)-a(n-4). G.f.: -x*(x^2-16*x-1)/((x^2-6*x+1)*(x^2+6*x+1)). [Colin Barker, Jul 16 2012]
Comments