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A041011
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Denominators of continued fraction convergents to sqrt(8).
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8
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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
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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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).
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)
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
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MAPLE
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with(combinat): a := n -> fibonacci(n + 1, 2)/2^(n mod 2):
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MATHEMATICA
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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 *)
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PROG
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(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
(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
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CROSSREFS
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KEYWORD
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nonn,cofr,frac,easy,changed
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AUTHOR
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EXTENSIONS
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First term 0 in b-file, formulas and programs removed by Georg Fischer, Jul 01 2019
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STATUS
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approved
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