A041364 -id:A041364 - cp's OEIS frontend

A040182 Continued fraction for sqrt(197).

14, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28

Offset: 0

N. J. A. Sloane

			14 + 1/(28 + 1/(28 + 1/(28 + 1/(28 + ...)))) = sqrt(197).

Cf. A041364/A041365 (convergents).

Cf. A040000, A040002, A040042.

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[197],300] (* Vladimir Joseph Stephan Orlovsky, Mar 28 2011 *)
PadRight[{14},120,{28}] (* Harvey P. Dale, May 02 2022 *)

From Elmo R. Oliveira, Feb 13 2024: (Start)
a(n) = 28 for n >= 1.
G.f.: 14*(1+x)/(1-x).
E.g.f.: 28*exp(x) - 14.
a(n) = 14*A040000(n) = 7*A040002(n) = 2*A040042(n). (End)

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

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

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)

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).

Cf. A041364, A040182.

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

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 *)

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)

Additional term from Colin Barker, Nov 16 2013

cp's OEIS Frontend

A040182 Continued fraction for sqrt(197).

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