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A041025
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Denominators of continued fraction convergents to sqrt(17).
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23
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1, 8, 65, 528, 4289, 34840, 283009, 2298912, 18674305, 151693352, 1232221121, 10009462320, 81307919681, 660472819768, 5365090477825, 43581196642368, 354014663616769, 2875698505576520, 23359602708228929, 189752520171407952, 1541379764079492545
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OFFSET
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0,2
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COMMENTS
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a(2*n+1) with b(2*n+1) := A041024(2*n+1), n>=0, give all (positive integer) solutions to Pell equation b^2 - 17*a^2 = +1, a(2*n) with b(2*n) := A041024(2*n), n>=0, give all (positive integer) solutions to Pell equation b^2 - 17*a^2 = -1 (cf. Emerson reference).
Bisection: a(2*n)= T(2*n+1,sqrt(17))/sqrt(17)= A078988(n), n>=0 and a(2*n+1)=8*S(n-1,66),n>=0, with T(n,x), resp. S(n,x), Chebyshev's polynomials of the first, resp. second kind. S(-1,x)=0. See A053120, resp. A049310. - Wolfdieter Lang, Jan 10 2003
Sqrt(17) = 8/2 + 8/65 + 8/(65*4289) + 8/(4289*283009) + ... . - Gary W. Adamson, Dec 26 2007
a(p) == ((p-1)/2)) mod p for odd primes p. - Gary W. Adamson, Feb 22 2009
For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 8's along the main diagonal and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
De Moivre's formula: a(n) = (r^n-s^n)/(r-s), for r>s, gives sequences with integers if r and s are conjugates. With r=4+sqrt(17) and s=4-sqrt(17), a(n+1)/a(n) converges to r=4+sqrt(17). - Sture Sjöstedt, Nov 11 2011
a(n) equals the number of words of length n on alphabet {0,1,...,8} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. Birmajer, J. B. Gil, M. D. Weiner, n the Enumeration of Restricted Words over a Finite Alphabet , J. Int. Seq. 19 (2016) # 16.1.3, Example 8
E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242, Thm. 1, p. 233.
Sergio Falcón and Ángel Plaza, The k-Fibonacci sequence and the Pascal 2-triangle, Chaos, Solitons & Fractals 2007; 33(1): 38-49.
S. Falcón & Á. Plaza, On k-Fibonacci sequences and polynomials and their derivatives, Chaos, Solitons & Fractals (2007).
M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Tanya Khovanova, Recursive Sequences
Kai Wang, On k-Fibonacci Sequences And Infinite Series List of Results and Examples, 2020.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (8,1).
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FORMULA
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G.f.: 1/(1 - 8*x - x^2).
a(n) = ((-i)^n)*S(n, 8*i), with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind and i^2 = -1. See A049310.
a(n) = F(n, 8), the n-th Fibonacci polynomial evaluated at x=8. - T. D. Noe, Jan 19 2006
a(n) = ((4 + sqrt(17))^n - (4 - sqrt(17))^n)/(2*sqrt(17)); a(n) = sum_{i=0..floor((n-1)/2)} binomial(n-1-i,i)*8^(n-1-2i). - Sergio Falcon, Sep 24 2007
Let T = the 2 X 2 matrix [0, 1; 1, 8]. Then T^n * [1, 0] = [a(n-2), a(n-1)]. - Gary W. Adamson, Dec 26 2007
a(n) = 8*a(n-1) + a(n-2), n>1; a(0)=1, a(1)=8. - Philippe Deléham, Nov 20 2008
sum_{n>=0} (-1)^n/(a(n)*a(n+1)) = sqrt(17)-4. - Vladimir Shevelev, Feb 23 2013
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MATHEMATICA
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a=0; lst={}; s=0; Do[a=s-(a-1); AppendTo[lst, a]; s+=a*8, {n, 3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)
CoefficientList[Series[1/(-z^2 - 8 z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[17], 30]] (* Harvey P. Dale, Aug 15 2011 *)
LinearRecurrence[{8, 1}, {1, 8}, 50] (* Sture Sjöstedt, Nov 11 2011 *)
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PROG
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(Sage) [lucas_number1(n, 8, -1) for n in range(1, 20)] # Zerinvary Lajos, Apr 25 2009
(MAGMA) I:=[1, 8]; [n le 2 select I[n] else 8*Self(n-1)+Self(n-2): n in [1..25]]; // Vincenzo Librandi, Feb 23 2013
(PARI) Vec(1/(1-8*x-x^2)+O(x^99)) \\ Charles R Greathouse IV, Dec 09 2014
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CROSSREFS
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Cf. A041024, A000045, A000129, A006190, A001076, A052918, A005668, A054413, A243399.
Sequence in context: A033118 A033126 A022039 * A163459 A081190 A189431
Adjacent sequences: A041022 A041023 A041024 * A041026 A041027 A041028
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KEYWORD
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nonn,cofr,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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