A041427 Denominators of continued fraction convergents to sqrt(229).

1, 7, 8, 15, 113, 3405, 23948, 27353, 51301, 386460, 11645101, 81902167, 93547268, 175449435, 1321693313, 39826248825, 280105435088, 319931683913, 600037119001, 4520191516920, 136205782626601, 957960669903127, 1094166452529728, 2052127122432855

Offset

0,2

Comments

The a(n) terms of this sequence can be constructed with the terms of sequence A154597. For the terms of the periodical sequence of the continued fraction for sqrt(229) see A040213. 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, 3420, 0, 0, 0, 0, 1).

Formula

a(5*n) = A154597(3*n+1), a(5*n+1) = (A154597(3*n+2) - A154597(3*n+1))/2, a(5*n+2) = (A154597(3*n+2) + A154597(3*n+1))/2, a(5*n+3) = A154597(3*n+2) and a(5*n+4) = A154597(3*n+3)/2. - Johannes W. Meijer, Jun 12 2010

G.f.: -(x^8 -7*x^7 +8*x^6 -15*x^5 +113*x^4 +15*x^3 +8*x^2 +7*x +1) / (x^10 +3420*x^5 -1). - Colin Barker, Nov 12 2013

a(n) = 3420*a(n-5) + a(n-10) for n>9. - Vincenzo Librandi, Dec 17 2013

Mathematica

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[229], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[229], 30]] (* Vincenzo Librandi, Dec 17 2013 *)
LinearRecurrence[{0, 0, 0, 0, 3420, 0, 0, 0, 0, 1}, {1, 7, 8, 15, 113, 3405, 23948, 27353, 51301, 386460}, 30] (* Harvey P. Dale, Oct 14 2020 *)

Prog

(Magma) I:=[1, 7, 8, 15, 113, 3405, 23948, 27353, 51301, 386460]; [n le 10 select I[n] else 3420*Self(n-5)+Self(n-10): n in [1..40]]; // Vincenzo Librandi, Dec 17 2013

Crossrefs

Cf. A041426, A166125, A040213, A041019, A041047, A041091, A041151, A041227, A041319, A041427 and A041551.

Sequence in context: A084377 A041675 A041098 * A042755 A093083 A323418

Adjacent sequences: A041424 A041425 A041426 * A041428 A041429 A041430

Keyword

nonn,easy,frac

Author

N. J. A. Sloane.

Status

approved