A041227 Denominators of continued fraction convergents to sqrt(125).

1, 5, 6, 11, 61, 1353, 6826, 8179, 15005, 83204, 1845493, 9310669, 11156162, 20466831, 113490317, 2517253805, 12699759342, 15217013147, 27916772489, 154800875592, 3433536035513, 17322481053157, 20756017088670, 38078498141827, 211148507797805

Offset

0,2

Comments

The a(n) terms of this sequence can be constructed with the terms of sequence A049666. For the terms of the periodical sequence of the continued fraction for sqrt(125) see A010186. We observe that its period is five. - Johannes W. Meijer, Jun 12 2010

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1364,0,0,0,0,1).

Formula

a(5*n) = A049666(3*n+1), a(5*n+1) = (A049666(3*n+2) - A049666(3*n+1))/2, a(5*n+2) = (A049666(3*n+2)+A049666(3*n+1))/2, a(5*n+3):= A049666(3*n+2) and a(5*n+4) = A049666(3*n+3)/2. - Johannes W. Meijer, Jun 12 2010

G.f.: -(x^8 -5*x^7 +6*x^6 -11*x^5 +61*x^4 +11*x^3 +6*x^2 +5*x +1) / ((x^2 +4*x -1)*(x^4 -7*x^3 +19*x^2 -3*x +1)*(x^4 +3*x^3 +19*x^2 +7*x +1)). - Colin Barker, Nov 12 2013

a(n) = 1364*a(n-5) + a(n-10). - Vincenzo Librandi, Dec 13 2013

Mathematica

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[125], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[125], 30]]  (* Vincenzo Librandi, Dec 13 2013 *)
LinearRecurrence[{0, 0, 0, 0, 1364, 0, 0, 0, 0, 1}, {1, 5, 6, 11, 61, 1353, 6826, 8179, 15005, 83204}, 30] (* Harvey P. Dale, Apr 29 2022 *)

Prog

(Magma) I:=[1, 5, 6, 11, 61, 1353, 6826, 8179, 15005, 83204]; [n le 10 select I[n] else 1364*Self(n-5)+Self(n-10): n in [1..40]]; // Vincenzo Librandi, Dec 13 2013

Crossrefs

Cf. A041226, A010186, A041019, A041047, A041091, A041151, A041227, A041319, A041427, A041551.

Sequence in context: A041355 A041050 A332523 * A042183 A362020 A328544

Adjacent sequences: A041224 A041225 A041226 * A041228 A041229 A041230

Keyword

nonn,frac,easy

Author

N. J. A. Sloane.

Status

approved