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A041113 Denominators of continued fraction convergents to sqrt(65). 9

%I

%S 1,16,257,4128,66305,1065008,17106433,274767936,4413393409,

%T 70889062480,1138638393089,18289103351904,293764292023553,

%U 4718517775728752,75790048703683585,1217359297034666112,19553538801258341377,314073980117168128144,5044737220675948391681

%N Denominators of continued fraction convergents to sqrt(65).

%C Sqrt(65) = 16/2 + 16/257 + 16/(257*66305) + 16/(66305*17106433) + ... - _Gary W. Adamson_, Jun 13 2008

%C 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]

%C 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

%H Nathaniel Johnston, <a href="/A041113/b041113.txt">Table of n, a(n) for n = 0..500</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16,1).

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

%F 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]

%F a(n) = ((8+sqrt(65))^(n+1)-(8-sqrt(65))^(n+1))/(2*sqrt(65)). [_Rolf Pleisch_, May 14 2011]

%t 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 *)

%t Denominator[Convergents[Sqrt[65], 30]] (* _Vincenzo Librandi_, Dec 11 2013 *)

%o (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

%Y Cf. A041112, A040056, A020822.

%K nonn,cofr,frac,easy

%O 0,2

%A _N. J. A. Sloane_.

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Last modified November 21 14:18 EST 2019. Contains 329371 sequences. (Running on oeis4.)