|
%I #41 Jul 15 2023 08:35:11
%S 1,36,1297,46728,1683505,60652908,2185188193,78727427856,
%T 2836372591009,102188140704180,3681609437941489,132640127906597784,
%U 4778726214075461713,172166783834623219452,6202782944260511361985,223472352777213032250912
%N Denominators of continued fraction convergents to sqrt(325).
%C From _Michael A. Allen_, Jul 13 2023: (Start)
%C Also called the 36-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 36 kinds of squares available. (End)
%H Vincenzo Librandi, <a href="/A041613/b041613.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 (36,1).
%F a(n) = F(n, 36), the n-th Fibonacci polynomial evaluated at x=36. - _T. D. Noe_, Jan 19 2006
%F From _Philippe Deléham_, Nov 23 2008: (Start)
%F a(n) = 36*a(n-1) + a(n-2) for n > 1; a(0)=1, a(1)=36.
%F G.f.: 1/(1 - 36*x - x^2). (End)
%p with (combinat):seq(fibonacci(3*n,3)/10, n=1..15); # _Zerinvary Lajos_, Apr 20 2008
%t a=0;lst={};s=0;Do[a=s-(a-1);AppendTo[lst,a];s+=a*36,{n,3*4!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Oct 27 2009 *)
%t Denominator[Convergents[Sqrt[325], 30]] (* _Vincenzo Librandi_ Dec 21 2013 *)
%Y Cf. A041612, A040306.
%Y Row n=36 of A073133, A172236 and A352361 and column k=36 of A157103.
%K nonn,frac,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Colin Barker_, Nov 20 2013
| 