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%I M4227 #152 Jun 03 2023 23:57:21
%S 0,1,6,37,228,1405,8658,53353,328776,2026009,12484830,76934989,
%T 474094764,2921503573,18003116202,110940200785,683644320912,
%U 4212806126257,25960481078454,159975692596981,985814636660340,6074863512559021,37434995712014466,230684837784645817
%N Denominators of continued fraction convergents to sqrt(10).
%C a(2*n+1) with b(2*n+1) := A005667(2*n+1), n >= 0, give all (positive integer) solutions to Pell equation b^2 - 10*a^2 = -1, a(2*n) with b(2*n) := A005667(2*n), n>=1, give all (positive integer) solutions to Pell equation b^2 - 10*a^2 = +1 (cf. Emerson reference).
%C Bisection: a(2*n)= 6*S(n-1,2*19) = 6*A078987(n-1), n >= 0 and a(2*n+1) = A097315(n), n >= 0, with S(n,x) Chebyshev's polynomials of the second kind. S(-1,x)=0. See A049310.
%C Sqrt(10) = 6/2 + 6/37 + 6/(37*1405) + 6/(1405*53353) + ... - _Gary W. Adamson_, Dec 21 2007
%C a(p) == 40^((p-1)/2) mod p, for odd primes p. - _Gary W. Adamson_, Feb 22 2009
%C For n>=2, a(n) equals the permanent of the (n-1)X(n-1) tridiagonal matrix with 6's along the main diagonal and 1's along the superdiagonal and the subdiagonal. - _John M. Campbell_, Jul 08 2011
%C For n>=1, a(n) equals the number of words of length n-1 on alphabet {0,1,...,6} avoiding runs of zeros of odd lengths. - _Milan Janjic_, Jan 28 2015
%C From _Michael A. Allen_, Feb 15 2023: (Start)
%C Also called the 6-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
%C a(n+1) 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 6 kinds of squares available. (End)
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A005668/b005668.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.2006.09.022">On the Fibonacci k-numbers</a>, Chaos, Solitons & Fractals 2007; 32(5): 1615-24.
%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 R. K. Guy, <a href="/A005667/a005667.pdf">Letter to N. J. A. Sloane, 1987</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=427">Encyclopedia of Combinatorial Structures 427</a>
%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 Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, and Fausto Jarquín-Zárate, <a href="https://arxiv.org/abs/1904.13002">The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d))</a>, arXiv:1904.13002 [math.NT], 2019.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%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/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,1).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F G.f.: x / (1 - 6*x - x^2).
%F a(n) = 6*a(n-1) + a(n-2).
%F a(n) = ((-i)^(n-1))*S(n-1, 3*i) with S(n, x) Chebyshev's polynomials of the second kind (see A049310) and i^2=-1.
%F a(n) = F(n, 6), the n-th Fibonacci polynomial evaluated at x=6. - _T. D. Noe_, Jan 19 2006
%F From _Sergio Falcon_, Sep 24 2007: (Start)
%F a(n) = ((3+sqrt(10))^n - (3-sqrt(10))^n)/(2*sqrt(10)).
%F a(n) = Sum_{i=0..floor((n-1)/2)} binomial(n-1-i,i)*6^(n-1-2*i). (End)
%F Sum_{n>=1}(-1)^(n-1)/(a(n)*a(n+1)) = sqrt(10) - 3. - _Vladimir Shevelev_, Feb 23 2013
%F a(n) = [M^(n+1)]_{0,0}, where M = [0,1; 1,6]. - _L. Edson Jeffery_, Aug 28 2013
%F a(-n) = -(-1)^n * a(n). - _Michael Somos_, May 28 2014
%F a(n) = 6^(n-1)*hypergeom([1-n/2, (1-n)/2], [1-n], -1/9)) for n >= 2. - _Peter Luschny_, Jun 28 2017
%e G.f. = x + 6*x^2 + 37*x^3 + 228*x^4 + 1405*x^5 + 8658*x^6 + 53353*x^7 + ...
%p evalf(sqrt(10),200); convert(%,confrac,fractionlist); fractionlist;
%p A005668:=-z/(-1+6*z+z**2); - _Simon Plouffe_ in his 1992 dissertation.
%p a := n -> `if`(n<2,n,6^(n-1)*hypergeom([1-n/2,(1-n)/2], [1-n], -1/9)):
%p seq(simplify(a(n)), n=0..23); # _Peter Luschny_, Jun 28 2017
%t LinearRecurrence[{6,1}, {0,1}, 30] (* _Vincenzo Librandi_, Feb 23 2013 *)
%t a[ n_] := (-I)^(n - 1) ChebyshevU[ n - 1, 3 I]; (* _Michael Somos_, May 28 2014 *)
%t a[ n_] := MatrixPower[ {{0, 1}, {1, 6}}, n + 1][[1, 1]]; (* _Michael Somos_, May 28 2014 *)
%t Fibonacci[Range[0,30],6] (* _G. C. Greubel_, Jun 06 2019 *)
%t Join[{0},Convergents[Sqrt[10],30]//Denominator] (* _Harvey P. Dale_, Dec 28 2022 *)
%o (Sage) from sage.combinat.sloane_functions import recur_gen3; it = recur_gen3(0,1,6,6,1,0); [next(it) for i in range(1,22)] # _Zerinvary Lajos_, Jul 09 2008
%o (Sage) [lucas_number1(n,6,-1) for n in range(0, 21)]# _Zerinvary Lajos_, Apr 24 2009
%o (Magma) [n le 2 select n-1 else 6*Self(n-1)+Self(n-2): n in [1..25]]; // _Vincenzo Librandi_, Feb 23 2013
%o (PARI) {a(n) = ([0, 1; 1, 6]^(n+1)) [1, 1]}; /* _Michael Somos_, May 28 2014 */
%o (PARI) {a(n) = (-I)^(n-1) * polchebyshev( n-1, 2, 3*I)}; /* _Michael Somos_, May 28 2014 */
%Y Row n=6 of A073133, A172236, A352361.
%Y Cf. A005667, A000045, A000129, A006190, A001076, A052918, A175185 (Pisano periods), A243399.
%K nonn,cofr,easy
%O 0,3
%A _N. J. A. Sloane_, _Simon Plouffe_, _R. K. Guy_
%E Chebyshev comments from _Wolfdieter Lang_, Jan 21 2003
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