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 A041011 Denominators of continued fraction convergents to sqrt(8). 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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). 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 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 *) PROG (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) x='x+O('x^99); concat(0, Vec((1+x-x^2)/(1-6*x^2+x^4))) \\ Altug Alkan, Mar 27 2016 CROSSREFS Cf. A000129, A010466, A041010. A089499, A136211, A152118. Sequence in context: A320047 A249221 A127040 * A152118 A041056 A042643 Adjacent sequences: A041008 A041009 A041010 * A041012 A041013 A041014 KEYWORD nonn,cofr,easy,changed AUTHOR N. J. A. Sloane EXTENSIONS Entry improved by Michael Somos First term 0 in b-file, formulas and programs removed by Georg Fischer, Jul 01 2019 STATUS approved

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Last modified November 30 01:30 EST 2023. Contains 367452 sequences. (Running on oeis4.)