A041011 Denominators of continued fraction convergents to sqrt(8).

1, 1, 5, 6, 29, 35, 169, 204, 985, 1189, 5741, 6930, 33461, 40391, 195025, 235416, 1136689, 1372105, 6625109, 7997214, 38613965, 46611179, 225058681, 271669860, 1311738121, 1583407981, 7645370045, 9228778026, 44560482149

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• frac
• easy

Comments

Sqrt(8) = 2 + continued fraction [0; 1, 4, 1, 4, 1, 4, ...] = 4/2 + 4/5 + 4/(5*29) + 4/(29*169) + 4/(169*985) + ... - Gary W. Adamson, Dec 21 2007

This is the sequence of Lehmer numbers U_n(sqrt(R),Q) with the parameters R = 4 and Q = -1. It is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all natural numbers n and m. - Peter Bala, May 12 2014

Apparently the same as A152118(n). - Georg Fischer, Jul 01 2019

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..199 [shifted by _Georg Fischer_, Jul 01 2019]
• Paul Barry, Symmetric Third-Order Recurring Sequences, Chebyshev Polynomials, and Riordan Arrays, JIS 12 (2009) 09.8.6.
• Hongshen Chua, A Study of Second-Order Linear Recurrence Sequences via Continuants, J. Int. Seq. (2023) Vol. 26, Art. 23.8.8.
• J. L. Ramirez and F. Sirvent, A q-Analogue of the Bi-Periodic Fibonacci Sequence, J. Int. Seq. 19 (2016) # 16.4.6, t_n at a=1, b=4.
• Eric Weisstein's World of Mathematics, Lehmer Number
• Index to divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Crossrefs

Cf. A000129, A010466, A041010. A089499, A136211, A152118.

Programs

Magma

I:=[1, 1, 5, 6]; [n le 4 select I[n] else 6*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 10 2013

Maple

with(combinat): a := n -> fibonacci(n + 1, 2)/2^(n mod 2):
seq(a(n), n = 0 .. 28); # Miles Wilson, Aug 04 2024

Mathematica

Denominator[NestList[(4/(4 + #))&, 0, 60]] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2010 *)
CoefficientList[Series[(x + x^2 - x^3)/(1 - 6 x^2 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 10 2013 *)
a0[n_] := ((3-2*Sqrt[2])^n*(2+Sqrt[2])-(-2+Sqrt[2])*(3+2*Sqrt[2])^n)/4 // Simplify
a1[n_] := (-(3-2*Sqrt[2])^n+(3+2*Sqrt[2])^n)/(4*Sqrt[2]) // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &,Range[20]]] (* Gerry Martens, Jul 11 2015 *)
LinearRecurrence[{0,6,0,-1},{1,1,5,6},40] (* Harvey P. Dale, Oct 21 2024 *)

PARI

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,0,6,0]^n*[1;1;5;6])[1,1] \\ Charles R Greathouse IV, Nov 13 2015

PARI

my(x='x+O('x^99)); concat(0, Vec((1+x-x^2)/(1-6*x^2+x^4))) \\ Altug Alkan, Mar 27 2016

Formula

a(2n) = A000129(2n+1), a(2n+1) = A000129(2n+2)/2.

a(n) = 6*a(n-2) - a(n-4). Also:

a(2n) = a(2n-1)+a(2n-2), a(2n+1)=4*a(2n)+a(2n-1).

G.f.: (1+x-x^2)/(1-6*x^2+x^4).

From Peter Bala, May 12 2014: (Start)

For n even, a(n) = (alpha^n - beta^n)/(alpha - beta), and for n odd, a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2), where alpha = 1 + sqrt(2) and beta = 1 - sqrt(2).

a(n) = Product_{k = 1..floor((n-1)/2)} ( 4 + 4*cos^2(k*Pi/n) ) for n >= 1. (End)

From Gerry Martens, Jul 11 2015: (Start)

Interspersion of 2 sequences [a0(n),a1(n)] for n>0:

a0(n) = ((3-2*sqrt(2))^n*(2+sqrt(2))-(-2+sqrt(2))*(3+2*sqrt(2))^n)/4.

a1(n) = (-(3-2*sqrt(2))^n+(3+2*sqrt(2))^n)/(4*sqrt(2)). (End)

a(n) = ((-(-1 - sqrt(2))^n - 3*(1-sqrt(2))^n + (-1+sqrt(2))^n + 3*(1+sqrt(2))^n))/(8*sqrt(2)). - Colin Barker, Mar 27 2016

Extensions

Entry improved by Michael Somos

First term 0 in b-file, formulas and programs removed by Georg Fischer, Jul 01 2019