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%I #66 May 01 2023 10:01:13
%S 1,16,257,4128,66305,1065008,17106433,274767936,4413393409,
%T 70889062480,1138638393089,18289103351904,293764292023553,
%U 4718517775728752,75790048703683585,1217359297034666112,19553538801258341377,314073980117168128144,5044737220675948391681,81029869510932342395040
%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
%C From _Michael A. Allen_, May 01 2023: (Start)
%C Also called the 16-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
%C a(n) 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 16 kinds of squares available. (End)
%H Nathaniel Johnston, <a href="/A041113/b041113.txt">Table of n, a(n) for n = 0..500</a>
%H Michael A. Allen and Kenneth Edwards, <a href="https://www.fq.math.ca/Papers1/60-5/allen.pdf">Fence tiling derived identities involving the metallonacci numbers squared or cubed</a>, Fib. Q. 60:5 (2022) 5-17.
%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 From _Philippe Deléham_, Nov 21 2008: (Start)
%F a(n) = 16*a(n-1) + a(n-2) for n > 1; a(0) = 1, a(1) = 16.
%F G.f.: 1/(1 - 16*x - x^2). (End)
%F a(n) = ((8+sqrt(65))^(n+1) - (8-sqrt(65))^(n+1))/(2*sqrt(65)). - _Rolf Pleisch_, May 14 2011
%F E.g.f.: exp(8*x)*(cosh(sqrt(65)*x) + 8*sinh(sqrt(65)*x)/sqrt(65)). - _Stefano Spezia_, Oct 28 2022
%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. A010517, A020822, A040056, A041112.
%Y Row n=16 of A073133, A172236 and A352361 and column k=16 of A157103.
%K nonn,frac,easy
%O 0,2
%A _N. J. A. Sloane_
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