A041131 Denominators of continued fraction convergents to sqrt(74).
1, 1, 2, 3, 5, 83, 88, 171, 259, 430, 7139, 7569, 14708, 22277, 36985, 614037, 651022, 1265059, 1916081, 3181140, 52814321, 55995461, 108809782, 164805243, 273615025, 4542645643, 4816260668, 9358906311, 14175166979, 23534073290, 390720339619, 414254412909
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,86,0,0,0,0,1).
Programs
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Magma
I:=[1, 1, 2, 3, 5, 83, 88, 171, 259, 430]; [n le 10 select I[n] else 86*Self(n-5)+Self(n-10): n in [1..40]]; // Vincenzo Librandi, Dec 11 2013
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Mathematica
Denominator/@Convergents[Sqrt[74], 50] (* Vladimir Joseph Stephan Orlovsky, Jul 05 2011 *) CoefficientList[Series[-(x^4 - 3 x^3 + 4 x^2 - 2 x + 1) (x^4 + 2 x^3 + 4 x^2 + 3 x + 1)/(x^10 + 86 x^5 - 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *) LinearRecurrence[{0,0,0,0,86,0,0,0,0,1},{1,1,2,3,5,83,88,171,259,430},50] (* Harvey P. Dale, Nov 09 2017 *)
Formula
G.f.: -(x^4-3*x^3+4*x^2-2*x+1)*(x^4+2*x^3+4*x^2+3*x+1) / (x^10+86*x^5-1). - Colin Barker, Nov 13 2013
a(n) = 86*a(n-5) + a(n-10). - Vincenzo Librandi, Dec 11 2013