A005668 Denominators of continued fraction convergents to sqrt(10). (Formerly M4227)

0, 1, 6, 37, 228, 1405, 8658, 53353, 328776, 2026009, 12484830, 76934989, 474094764, 2921503573, 18003116202, 110940200785, 683644320912, 4212806126257, 25960481078454, 159975692596981, 985814636660340, 6074863512559021, 37434995712014466, 230684837784645817

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

Kai Wang, On k-Fibonacci Sequences And Infinite Series List of Results and Examples, 2020.

Index entries for linear recurrences with constant coefficients, signature (6,1).

Index entries for sequences related to Chebyshev polynomials.

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

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

Row n=6 of A073133, A172236, A352361.

Cf. A005667, A000045, A000129, A006190, A001076, A052918, A175185 (Pisano periods), A243399.

Sequence in context: A180032 A022035 A255119 * A018904 A192807 A076026

Adjacent sequences: A005665 A005666 A005667 * A005669 A005670 A005671

Keyword

nonn,cofr,easy

Author

N. J. A. Sloane, Simon Plouffe, R. K. Guy

Extensions

Chebyshev comments from Wolfdieter Lang, Jan 21 2003

Status

approved