OFFSET
0,3
COMMENTS
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).
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.
Sqrt(10) = 6/2 + 6/37 + 6/(37*1405) + 6/(1405*53353) + ... - Gary W. Adamson, Dec 21 2007
a(p) == 40^((p-1)/2) mod p, for odd primes p. - Gary W. Adamson, Feb 22 2009
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
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
From Michael A. Allen, Feb 15 2023: (Start)
Also called the 6-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
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)
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
D. Birmajer, J. B. Gil, and M. D. Weiner, On 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, On the Fibonacci k-numbers, Chaos, Solitons & Fractals 2007; 32(5): 1615-24.
Sergio Falcón and Ángel Plaza, The k-Fibonacci sequence and the Pascal 2-triangle Chaos, Solitons & Fractals 2007; 33(1): 38-49.
R. K. Guy, Letter to N. J. A. Sloane, 1987
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 427
Milan 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
Pablo Lam-Estrada, Myriam Rosalía Maldonado-Ramírez, José Luis López-Bonilla, and Fausto Jarquín-Zárate, The sequences of Fibonacci and Lucas for each real quadratic fields Q(Sqrt(d)), arXiv:1904.13002 [math.NT], 2019.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (6,1).
FORMULA
G.f.: x / (1 - 6*x - x^2).
a(n) = 6*a(n-1) + a(n-2).
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.
a(n) = F(n, 6), the n-th Fibonacci polynomial evaluated at x=6. - T. D. Noe, Jan 19 2006
From Sergio Falcon, Sep 24 2007: (Start)
a(n) = ((3+sqrt(10))^n - (3-sqrt(10))^n)/(2*sqrt(10)).
a(n) = Sum_{i=0..floor((n-1)/2)} binomial(n-1-i,i)*6^(n-1-2*i). (End)
Sum_{n>=1}(-1)^(n-1)/(a(n)*a(n+1)) = sqrt(10) - 3. - Vladimir Shevelev, Feb 23 2013
a(n) = [M^(n+1)]_{0,0}, where M = [0,1; 1,6]. - L. Edson Jeffery, Aug 28 2013
a(-n) = -(-1)^n * a(n). - Michael Somos, May 28 2014
a(n) = 6^(n-1)*hypergeom([1-n/2, (1-n)/2], [1-n], -1/9) for n >= 2. - Peter Luschny, Jun 28 2017
G.f.: x/(1 - 6*x - x^2) = Sum_{n >= 0} x^(n+1) *( Product_{k = 1..n} (m*k + 6 - m + x)/(1 + m*k*x) ) for arbitrary m (a telescoping series). - Peter Bala, May 08 2024
EXAMPLE
G.f. = x + 6*x^2 + 37*x^3 + 228*x^4 + 1405*x^5 + 8658*x^6 + 53353*x^7 + ...
MAPLE
evalf(sqrt(10), 200); convert(%, confrac, fractionlist); fractionlist;
A005668:=-z/(-1+6*z+z**2); - Simon Plouffe in his 1992 dissertation.
a := n -> `if`(n<2, n, 6^(n-1)*hypergeom([1-n/2, (1-n)/2], [1-n], -1/9)):
seq(simplify(a(n)), n=0..23); # Peter Luschny, Jun 28 2017
MATHEMATICA
LinearRecurrence[{6, 1}, {0, 1}, 30] (* Vincenzo Librandi, Feb 23 2013 *)
a[ n_] := (-I)^(n - 1) ChebyshevU[ n - 1, 3 I]; (* Michael Somos, May 28 2014 *)
a[ n_] := MatrixPower[ {{0, 1}, {1, 6}}, n + 1][[1, 1]]; (* Michael Somos, May 28 2014 *)
Fibonacci[Range[0, 30], 6] (* G. C. Greubel, Jun 06 2019 *)
Join[{0}, Convergents[Sqrt[10], 30]//Denominator] (* Harvey P. Dale, Dec 28 2022 *)
PROG
(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
(Sage) [lucas_number1(n, 6, -1) for n in range(0, 21)]# Zerinvary Lajos, Apr 24 2009
(Magma) [n le 2 select n-1 else 6*Self(n-1)+Self(n-2): n in [1..25]]; // Vincenzo Librandi, Feb 23 2013
(PARI) {a(n) = ([0, 1; 1, 6]^(n+1)) [1, 1]}; /* Michael Somos, May 28 2014 */
(PARI) {a(n) = (-I)^(n-1) * polchebyshev( n-1, 2, 3*I)}; /* Michael Somos, May 28 2014 */
CROSSREFS
KEYWORD
nonn,cofr,frac,easy
AUTHOR
EXTENSIONS
Chebyshev comments from Wolfdieter Lang, Jan 21 2003
STATUS
approved