|
%I #118 May 17 2024 10:19:16
%S 1,8,65,528,4289,34840,283009,2298912,18674305,151693352,1232221121,
%T 10009462320,81307919681,660472819768,5365090477825,43581196642368,
%U 354014663616769,2875698505576520,23359602708228929,189752520171407952,1541379764079492545
%N Denominators of continued fraction convergents to sqrt(17).
%C 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).
%C 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
%C Sqrt(17) = 8/2 + 8/65 + 8/(65*4289) + 8/(4289*283009) + ... . - _Gary W. Adamson_, Dec 26 2007
%C 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
%C 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
%C 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
%C From _Michael A. Allen_, Feb 21 2023: (Start)
%C Also called the 8-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
%C a(n) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 8 kinds of squares available. (End)
%H Vincenzo Librandi, <a href="/A041025/b041025.txt">Table of n, a(n) for n = 0..1000</a>
%H Michael A. Allen and Kenneth Edwards, <a href="https://www.fq.math.ca/Papers1/60-5/allen.pdf">Fence tiling derived identities involving the metallonacci numbers squared or cubed</a>, Fib. Q. 60:5 (2022) 5-17.
%H D. Birmajer, J. B. Gil, and M. D. Weiner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Gil/gil6.html">On the Enumeration of Restricted Words over a Finite Alphabet</a>, J. Int. Seq. 19 (2016) # 16.1.3, Example 8.
%H E. I. Emerson, <a href="http://www.fq.math.ca/Scanned/7-3/emerson.pdf">Recurrent Sequences in the Equation DQ^2=R^2+N</a>, Fib. Quart., 7 (1969), pp. 231-242, Thm. 1, p. 233.
%H Sergio Falcón and Ángel Plaza, <a href="http://dx.doi.org/10.1016/j.chaos.2006.10.022">The k-Fibonacci sequence and the Pascal 2-triangle</a>, Chaos, Solitons & Fractals 2007; 33(1): 38-49.
%H S. Falcón and Á. Plaza, <a href="http://dx.doi.org/10.1016/j.chaos.2007.03.007">On k-Fibonacci sequences and polynomials and their derivatives</a>, Chaos, Solitons & Fractals (2007).
%H Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Kai Wang, <a href="https://www.researchgate.net/publication/339487198_On_k-Fibonacci_Sequences_And_Infinite_Series_List_of_Results_and_Examples">On k-Fibonacci Sequences And Infinite Series List of Results and Examples</a>, 2020.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,1).
%F G.f.: 1/(1 - 8*x - x^2).
%F 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.
%F a(n) = F(n, 8), the n-th Fibonacci polynomial evaluated at x=8. - _T. D. Noe_, Jan 19 2006
%F From _Sergio Falcon_, Sep 24 2007: (Start)
%F a(n) = ((4 + sqrt(17))^n - (4 - sqrt(17))^n)/(2*sqrt(17));
%F a(n) = Sum_{i=0..floor((n-1)/2)} binomial(n-1-i,i)*8^(n-1-2i). (End)
%F Let T be 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
%F a(n) = 8*a(n-1) + a(n-2), n > 1; a(0)=1, a(1)=8. - _Philippe Deléham_, Nov 20 2008
%F a(p) == ((p-1)/2)) (mod p) for odd primes p. - _Gary W. Adamson_, Feb 22 2009
%F Sum_{n>=0} (-1)^n/(a(n)*a(n+1)) = sqrt(17) - 4. - _Vladimir Shevelev_, Feb 23 2013
%F G.f.: x/(1 - 8*x - x^2) = Sum_{n >= 0} x^n *( Product_{k = 1..n} (m*k + 8 - m + x)/(1 + m*k*x) ) for arbitrary m (a telescoping series). - _Peter Bala_, May 08 2024
%t CoefficientList[Series[1/(-z^2 - 8 z + 1), {z, 0, 200}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 23 2011 *)
%t Denominator[Convergents[Sqrt[17],30]] (* _Harvey P. Dale_, Aug 15 2011 *)
%t LinearRecurrence[{8,1}, {1,8}, 50] (* _Sture Sjöstedt_, Nov 11 2011 *)
%o (Sage) [lucas_number1(n,8,-1) for n in range(1, 20)] # _Zerinvary Lajos_, Apr 25 2009
%o (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
%o (PARI) Vec(1/(1-8*x-x^2)+O(x^99)) \\ _Charles R Greathouse IV_, Dec 09 2014
%Y Cf. A041024, A040012.
%Y Cf. A000045, A000129, A006190, A001076, A052918, A005668, A054413, A243399.
%Y Row n=8 of A073133, A172236 and A352361.
%Y Cf. A099369 (squares).
%K nonn,cofr,easy
%O 0,2
%A _N. J. A. Sloane_
| 