login
Denominators of continued fraction convergents to sqrt(101).
9

%I #56 Sep 30 2024 12:51:41

%S 1,20,401,8040,161201,3232060,64802401,1299280080,26050404001,

%T 522307360100,10472197606001,209966259480120,4209797387208401,

%U 84405914003648140,1692328077460171201,33930967463207072160,680311677341601614401

%N Denominators of continued fraction convergents to sqrt(101).

%C Generalized Pell numbers (A000129). Antidiagonals of A038207. - _Mark Dols_, Aug 31 2009

%C a(n) equals the number of words of length n on alphabet {0,1,...,20} 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 20-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 20 kinds of squares available. (End)

%H Vincenzo Librandi, <a href="/A041181/b041181.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 (20,1).

%F a(n) = Fibonacci(n+1, 20), the n-th Fibonacci polynomial evaluated at x=20. - _T. D. Noe_, Jan 19 2006

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

%F a(n) = 20*a(n-1) + a(n-2) for n>1, a(0)=1, a(1)=20.

%F G.f.: 1/(1-20*x-x^2). (End)

%t Denominator[Convergents[Sqrt[101], 30]] (* _Vincenzo Librandi_, Dec 12 2013 *)

%t LinearRecurrence[{20,1},{1,20},20] (* _Harvey P. Dale_, Mar 17 2020 *)

%o (Magma) [n le 2 select (20)^(n-1) else 20*Self(n-1)+Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Dec 12 2013

%o (SageMath)

%o A041181=BinaryRecurrenceSequence(20,1,1,20)

%o [A041181(n) for n in range(31)] # _G. C. Greubel_, Sep 29 2024

%Y Cf. A000129, A038207, A040090, A041180.

%Y Cf. similar sequences listed in A243399.

%Y Row n=20 of A073133, A172236 and A352361 and column k=20 of A157103.

%K nonn,frac,easy,less

%O 0,2

%A _N. J. A. Sloane_