A041153 Denominators of continued fraction convergents to sqrt(86).

1, 3, 4, 7, 11, 95, 106, 201, 307, 1122, 20503, 62631, 83134, 145765, 228899, 1976957, 2205856, 4182813, 6388669, 23348820, 426667429, 1303351107, 1730018536, 3033369643, 4763388179, 41140475075, 45903863254, 87044338329, 132948201583, 485888943078

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,20810,0,0,0,0,0,0,0,0,0,-1).

Crossrefs

Cf. A041152, A010159, A020843, A010537.

Programs

Magma

I:=[1, 3, 4, 7, 11, 95, 106, 201, 307, 1122, 20503, 62631, 83134, 145765, 228899, 1976957, 2205856, 4182813, 6388669, 23348820]; [n le 20 select I[n] else 20810*Self(n-10)-Self(n-20): n in [1..40]]; // Vincenzo Librandi, Dec 12 2013

Mathematica

Denominator/@Convergents[Sqrt[86], 50] (* Vladimir Joseph Stephan Orlovsky, Jul 05 2011 *)
CoefficientList[Series[-(x^18 - 3 x^17 + 4 x^16 - 7 x^15 + 11 x^14 - 95 x^13 + 106 x^12 - 201 x^11 + 307 x^10 - 1122 x^9 - 307 x^8 - 201 x^7 - 106 x^6 - 95 x^5 - 11 x^4 - 7 x^3 - 4 x^2 - 3 x - 1)/(x^20 - 20810 x^10 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 12 2013 *)

Formula

G.f.: -(x^18 -3*x^17 +4*x^16 -7*x^15 +11*x^14 -95*x^13 +106*x^12 -201*x^11 +307*x^10 -1122*x^9 -307*x^8 -201*x^7 -106*x^6 -95*x^5 -11*x^4 -7*x^3 -4*x^2 -3*x-1) / (x^20-20810*x^10+1). - Colin Barker, Nov 13 2013

a(n) = 20810*a(n-10) - a(n-20). - Vincenzo Librandi, Dec 12 2013