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A041113 Denominators of continued fraction convergents to sqrt(65). 6
1, 16, 257, 4128, 66305, 1065008, 17106433, 274767936, 4413393409, 70889062480, 1138638393089, 18289103351904, 293764292023553, 4718517775728752, 75790048703683585, 1217359297034666112, 19553538801258341377, 314073980117168128144, 5044737220675948391681 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Sqrt(65) = 16/2 + 16/257 + 16/(257*66305) + 16/(66305*17106433) + ... - Gary W. Adamson, Jun 13 2008

For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 16'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,...,16} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015

LINKS

Nathaniel Johnston, Table of n, a(n) for n = 0..500

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (16,1).

FORMULA

a(n) = F(n, 16), the n-th Fibonacci polynomial evaluated at x=16. - T. D. Noe, Jan 19 2006

a(n) = 16*a(n-1)+a(n-2) for n>1, a(0)=1, a(1)=16. G.f.: 1/(1-16*x-x^2). [Philippe Deléham, Nov 21 2008]

a(n) = ((8+sqrt(65))^(n+1)-(8-sqrt(65))^(n+1))/(2*sqrt(65)). [Rolf Pleisch, May 14 2011]

MATHEMATICA

a=0; lst={}; s=0; Do[a=s-(a-1); AppendTo[lst, a]; s+=a*16, {n, 3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)

Denominator[Convergents[Sqrt[65], 30]] (* Vincenzo Librandi, Dec 11 2013 *)

PROG

(MAGMA) I:=[1, 16]; [n le 2 select I[n] else 16*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 11 2013

CROSSREFS

Cf. A041112, A040056, A020822.

Sequence in context: A223433 A223593 A067223 * A041482 A220742 A266023

Adjacent sequences:  A041110 A041111 A041112 * A041114 A041115 A041116

KEYWORD

nonn,cofr,frac,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified June 26 23:21 EDT 2017. Contains 288777 sequences.