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 A041113 #73 Jan 05 2025 19:51:35
%S A041113 1,16,257,4128,66305,1065008,17106433,274767936,4413393409,
%T A041113 70889062480,1138638393089,18289103351904,293764292023553,
%U A041113 4718517775728752,75790048703683585,1217359297034666112,19553538801258341377,314073980117168128144,5044737220675948391681,81029869510932342395040
%N A041113 Denominators of continued fraction convergents to sqrt(65).
%C A041113 Sqrt(65) = 16/2 + 16/257 + 16/(257*66305) + 16/(66305*17106433) + ... - _Gary W. Adamson_, Jun 13 2008
%C A041113 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 A041113 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 A041113 From _Michael A. Allen_, May 01 2023: (Start)
%C A041113 Also called the 16-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
%C A041113 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 A041113 Nathaniel Johnston, <a href="/A041113/b041113.txt">Table of n, a(n) for n = 0..500</a>
%H A041113 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 A041113 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A041113 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16,1).
%F A041113 a(n) = Fibonacci(n+1, 16). - _T. D. Noe_, Jan 19 2006
%F A041113 From _Philippe Deléham_, Nov 21 2008: (Start)
%F A041113 a(n) = 16*a(n-1) + a(n-2) for n > 1; a(0) = 1, a(1) = 16.
%F A041113 G.f.: 1/(1 - 16*x - x^2). (End)
%F A041113 a(n) = ((8+sqrt(65))^(n+1) - (8-sqrt(65))^(n+1))/(2*sqrt(65)). - _Rolf Pleisch_, May 14 2011
%F A041113 E.g.f.: exp(8*x)*(cosh(sqrt(65)*x) + 8*sinh(sqrt(65)*x)/sqrt(65)). - _Stefano Spezia_, Oct 28 2022
%t A041113 Denominator[Convergents[Sqrt[65], 30]] (* _Vincenzo Librandi_, Dec 11 2013 *)
%t A041113 Fibonacci[Range[30], 16] (* _G. C. Greubel_, Sep 29 2024 *)
%o A041113 (Magma) [n le 2 select (16)^(n-1) else 16*Self(n-1)+Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Dec 11 2013
%o A041113 (SageMath)
%o A041113 A041113=BinaryRecurrenceSequence(16,1,1,16)
%o A041113 [A041113(n) for n in range(0,31)] # _G. C. Greubel_, Sep 29 2024
%Y A041113 Cf. A010517, A020822, A040056, A041112.
%Y A041113 Row n=16 of A073133, A172236 and A352361 and column k=16 of A157103.
%K A041113 nonn,frac,easy
%O A041113 0,2
%A A041113 _N. J. A. Sloane_