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A041025 Denominators of continued fraction convergents to sqrt(17). 17
1, 8, 65, 528, 4289, 34840, 283009, 2298912, 18674305, 151693352, 1232221121, 10009462320, 81307919681, 660472819768, 5365090477825, 43581196642368, 354014663616769, 2875698505576520, 23359602708228929, 189752520171407952, 1541379764079492545 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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 zeroes of odd lengths. - Milan Janjic, Jan 28 2015

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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, 2014; http://matinf.pmfbl.org/wp-content/uploads/2015/01/za-arhiv-18.-1.pdf

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (8,1).

FORMULA

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

MATHEMATICA

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 *)

PROG

(Sage) [lucas_number1(n, 8, -1) for n in xrange(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

CROSSREFS

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

KEYWORD

nonn,cofr,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 26 15:40 EDT 2017. Contains 285446 sequences.