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A041427
Denominators of continued fraction convergents to sqrt(229).
10
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
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
KEYWORD
nonn,easy,frac
AUTHOR
STATUS
approved