This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A041145 #55 Jan 05 2025 19:51:35
%S A041145 1,18,325,5868,105949,1912950,34539049,623615832,11259624025,
%T A041145 203296848282,3670602893101,66274148924100,1196605283526901,
%U A041145 21605169252408318,390089651826876625,7043218902136187568,127168029890278252849,2296067756927144738850,41456387654578883552149
%N A041145 Denominators of continued fraction convergents to sqrt(82).
%C A041145 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 A041145 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 A041145 From _Michael A. Allen_, May 03 2023: (Start)
%C A041145 Also called the 18-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
%C A041145 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 A041145 Vincenzo Librandi, <a href="/A041145/b041145.txt">Table of n, a(n) for n = 0..200</a>
%H A041145 Michael A. Allen and Kenneth Edwards, <a href="https://web.archive.org/web/2024*/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 A041145 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H A041145 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (18,1).
%F A041145 a(n) = Fibonacci(n+1, 18), the n-th Fibonacci polynomial evaluated at x=18. - _T. D. Noe_, Jan 19 2006
%F A041145 From _Philippe Deléham_, Nov 21 2008: (Start)
%F A041145 a(n) = 18*a(n-1) + a(n-2) for n > 1; a(0)=1, a(1)=18.
%F A041145 G.f.: 1/(1 - 18*x - x^2). (End)
%F A041145 E.g.f.: exp(9*x)*(cosh(sqrt(82)*x) + 9*sinh(sqrt(82)*x)/sqrt(82)). - _Stefano Spezia_, Oct 02 2024
%t A041145 Denominator[Convergents[Sqrt[82], 30]] (* _Vincenzo Librandi_, Dec 11 2013 *)
%t A041145 Fibonacci[Range[30], 18] (* _G. C. Greubel_, Sep 29 2024 *)
%o A041145 (Magma)
%o A041145 [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 A041145 (SageMath)
%o A041145 A041145=BinaryRecurrenceSequence(18,1,1,18)
%o A041145 [A041145(n) for n in range(31)] # _G. C. Greubel_, Sep 29 2024
%Y A041145 Cf. A010533, A020839, A040072, A041144.
%Y A041145 Cf. similar sequences listed in A243399.
%Y A041145 Row n=18 of A073133, A172236 and A352361 and column k=18 of A157103.
%K A041145 nonn,frac,easy
%O A041145 0,2
%A A041145 _N. J. A. Sloane_