%I #54 Oct 03 2024 03:52:09

%S 1,18,325,5868,105949,1912950,34539049,623615832,11259624025,

%T 203296848282,3670602893101,66274148924100,1196605283526901,

%U 21605169252408318,390089651826876625,7043218902136187568,127168029890278252849,2296067756927144738850,41456387654578883552149

%N Denominators of continued fraction convergents to sqrt(82).

%C For n >= 2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 18'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,...,18} avoiding runs of zeros of odd lengths. - _Milan Janjic_, Jan 28 2015

%C From _Michael A. Allen_, May 03 2023: (Start)

%C Also called the 18-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 18 kinds of squares available. (End)

%H Vincenzo Librandi, <a href="/A041145/b041145.txt">Table of n, a(n) for n = 0..200</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 (18,1).

%F a(n) = Fibonacci(n+1, 18), the n-th Fibonacci polynomial evaluated at x=18. - _T. D. Noe_, Jan 19 2006

%F From _Philippe Deléham_, Nov 21 2008: (Start)

%F a(n) = 18*a(n-1) + a(n-2) for n > 1; a(0)=1, a(1)=18.

%F G.f.: 1/(1 - 18*x - x^2). (End)

%F E.g.f.: exp(9*x)*(cosh(sqrt(82)*x) + 9*sinh(sqrt(82)*x)/sqrt(82)). - _Stefano Spezia_, Oct 02 2024

%t Denominator[Convergents[Sqrt[82], 30]] (* _Vincenzo Librandi_, Dec 11 2013 *)

%t Fibonacci[Range[30], 18] (* _G. C. Greubel_, Sep 29 2024 *)

%o (Magma)

%o [n le 2 select (18)^(n-1) else 18*Self(n-1)+Self(n-2): n in [1..30]]; // _G. C. Greubel_, Sep 29 2024

%o (SageMath)

%o A041145=BinaryRecurrenceSequence(18,1,1,18)

%o [A041145(n) for n in range(31)] # _G. C. Greubel_, Sep 29 2024

%Y Cf. A010533, A020839, A040072, A041144.

%Y Cf. similar sequences listed in A243399.

%Y Row n=18 of A073133, A172236 and A352361 and column k=18 of A157103.

%K nonn,frac,easy

%O 0,2

%A _N. J. A. Sloane_