A041063 Denominators of continued fraction convergents to sqrt(38).
1, 6, 73, 444, 5401, 32850, 399601, 2430456, 29565073, 179820894, 2187415801, 13304315700, 161839204201, 984339540906, 11973913695073, 72827821711344, 885907774231201, 5388274467098550, 65545201379413801, 398659482743581356, 4849458994302390073
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Index entries for linear recurrences with constant coefficients, signature (0,74,0,-1).
Programs
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Magma
I:=[1, 6, 73, 444]; [n le 4 select I[n] else 74*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 10 2013
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Mathematica
Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[38], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011*) Denominator[Convergents[Sqrt[38], 30]] (* Vincenzo Librandi, Dec 10 2013 *) a0[n_] := ((38+6*Sqrt[38])/(37+6*Sqrt[38])^n+(38-6*Sqrt[38])*(37+6*Sqrt[38])^n)/76 // Simplify a1[n_] := (-1/(37+6*Sqrt[38])^n+(37+6*Sqrt[38])^n)/(2*Sqrt[38]) // FullSimplify Flatten[MapIndexed[{a0[#],a1[#]}&, Range[20]]] (* Gerry Martens, Jul 11 2015 *) LinearRecurrence[{0,74,0,-1},{1,6,73,444},30] (* Harvey P. Dale, Feb 29 2024 *)
Formula
G.f.: -(x^2-6*x-1) / (x^4-74*x^2+1). - Colin Barker, Nov 12 2013
a(n) = 74*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 10 2013
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
a0(n) = ((38+6*sqrt(38))/(37+6*sqrt(38))^n+(38-6*sqrt(38))*(37+6*sqrt(38))^n)/76;
a1(n) = (-1/(37+6*sqrt(38))^n+(37+6*sqrt(38))^n)/(2*sqrt(38)). (End)
Extensions
More terms from Colin Barker, Nov 12 2013