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A041041 Denominators of continued fraction convergents to sqrt(26). 13
1, 10, 101, 1020, 10301, 104030, 1050601, 10610040, 107151001, 1082120050, 10928351501, 110365635060, 1114584702101, 11256212656070, 113676711262801, 1148023325284080, 11593909964103601, 117087122966320090, 1182465139627304501, 11941738519239365100 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Generalized Fibonacci sequence.

Sqrt(26) = 10/2 + 10/101 + 10/(101*10301) + 10/(10301*1050601) + ... - Gary W. Adamson, Jun 13 2008

For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 10's along the main diagonal and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011

a(n) equals the number of words of length n on alphabet {0,1,...,10} avoiding runs of zeroes of odd lengths. - Milan Janjic, Jan 28 2015

REFERENCES

S. Falcon & A. Plaza: The k-Fibonacci sequence and the Pascal 2-triangle, Chaos, Solitons & Fractals, 33 (2007)

S. Falcon & A. Plaza: On k-Fibonacci sequences and polynomials and their derivatives, Chaos, Solitons & Fractals (2007)

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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.

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

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (10,1).

FORMULA

a(n) = 10*a(n-1) + a(n-2), n>=1; a(-1):=0, a(0)=1.

G.f.: 1/(1-10*x-x^2).

a(n) = S(n, 10*I)*(-I)^n with I^2:=-1 and S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind. See A049310.

a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = 5+sqrt(26), am = 5-sqrt(26) = -1/ap.

a(n) = F(n, 10), the n-th Fibonacci polynomial evaluated at x=10. - T. D. Noe, Jan 19 2006

a(n) = ((5+Sqrt[26])^n-(5-Sqrt[26])^n)/(2Sqrt[26]); a(n) = Sum[Binomial[n-1-i,i]*10^{n-1-2i}, {i,0,Floor[(n-1)/2]}]. - Sergio Falcon, Sep 24 2007

MATHEMATICA

a=0; lst={}; s=0; Do[a=s-(a-1); AppendTo[lst, a]; s+=a*10, {n, 3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)

Denominator[Convergents[Sqrt[26], 30]] (* Vincenzo Librandi, Dec 10 2013 *)

PROG

(Sage) [lucas_number1(n, 10, -1) for n in xrange(1, 19)] # Zerinvary Lajos, Apr 26 2009

CROSSREFS

Cf. A041040.

Cf. squares A099374.

Cf. A000045, A000129, A006190, A001076, A052918, A005668, A054413, A041025, A099371.

Cf. A243399.

Sequence in context: A180175 A267526 A261199 * A163461 A081192 A279174

Adjacent sequences:  A041038 A041039 A041040 * A041042 A041043 A041044

KEYWORD

nonn,cofr,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Extended by T. D. Noe, May 23 2011

STATUS

approved

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Last modified January 23 21:40 EST 2017. Contains 281216 sequences.