A041365 Denominators of continued fraction convergents to sqrt(197).

1, 28, 785, 22008, 617009, 17298260, 484968289, 13596410352, 381184458145, 10686761238412, 299610499133681, 8399780736981480, 235493471134615121, 6602216972506204868, 185097568701308351425, 5189334140609140044768

Offset

0,2

Comments

From Michael A. Allen, May 16 2023: (Start)

Also called the 28-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.

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 28 kinds of squares available. (End)

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (28,1).

Formula

a(n) = F(n, 28), the n-th Fibonacci polynomial evaluated at x=28. - T. D. Noe, Jan 19 2006

From Philippe Deléham, Nov 21 2008: (Start)

a(n) = 28*a(n-1) + a(n-2), n > 1; a(0)= 1, a(1)=28.

G.f.: 1/(1-28*x-x^2). (End)

Mathematica

a=0; lst={}; s=0; Do[a=s-(a-1); AppendTo[lst, a]; s+=a*28, {n, 3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)
Denominator[Convergents[Sqrt[197], 30]] (* Vincenzo Librandi, Dec 16 2013 *)
LinearRecurrence[{28, 1}, {1, 28}, 20] (* Harvey P. Dale, Mar 07 2021 *)

Crossrefs

Cf. A041364, A040182.

Row n=28 of A073133, A172236 and A352361 and column k=28 of A157103.

Sequence in context: A267732 A009972 A114037 * A042514 A158659 A214133

Adjacent sequences: A041362 A041363 A041364 * A041366 A041367 A041368

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Extensions

Additional term from Colin Barker, Nov 16 2013

Status

approved