login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A005668 Denominators of continued fraction convergents to sqrt(10).
(Formerly M4227)
27
0, 1, 6, 37, 228, 1405, 8658, 53353, 328776, 2026009, 12484830, 76934989, 474094764, 2921503573, 18003116202, 110940200785, 683644320912, 4212806126257, 25960481078454, 159975692596981, 985814636660340 (list; graph; refs; listen; history; text; internal format)
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 zeroes of odd lengths. - Milan Janjic, Jan 28 2015

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

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 Falcon and Ángel Plaza, On the Fibonacci k-numbers, Chaos, Solitons & Fractals 2007; 32(5): 1615-24.

Sergio Falcon and Ángel Plaza, The k-Fibonacci sequence and the Pascal 2-triangle Chaos, Solitons & Fractals 2007; 33(1): 38-49.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 427

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

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

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) = 6a(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

a(n) = ((3+Sqrt[10])^n-(3-Sqrt[10])^n)/(2Sqrt[10]); a(n) = Sum_0^{Floor[(n-1)/2]} Binomial[n-1-i,i]*6^{n-1-2i}. - Sergio Falcon, Sep 24 2007

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

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.

MATHEMATICA

a = 0; lst = {a}; s = 0; Do[a = s - (a - 1); AppendTo[lst, a]; s += 6 a, {n, 3 4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)

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 *)

PROG

(Sage) from sage.combinat.sloane_functions import recur_gen3; it = recur_gen3(0, 1, 6, 6, 1, 0); [it.next() for i in xrange(1, 22)] # Zerinvary Lajos, Jul 09 2008

(Sage) [lucas_number1(n, 6, -1) for n in xrange(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

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

Cf. A243399.

Sequence in context: A180032 A022035 A255119 * A018904 A192807 A076026

Adjacent sequences:  A005665 A005666 A005667 * A005669 A005670 A005671

KEYWORD

nonn,cofr,easy,changed

AUTHOR

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

EXTENSIONS

Chebyshev comments from Wolfdieter Lang, Jan 21 2003

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

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