A041312 -id:A041312 - cp's OEIS frontend

A040156 Continued fraction for sqrt(170).

13, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26, 26

Offset: 0

N. J. A. Sloane

			13 + 1/(26 + 1/(26 + 1/(26 + 1/(26 + ...)))) = sqrt(170).

Cf. A040000, A041312/A041313 (convergents).

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

ContinuedFraction[Sqrt[170],300] (* Vladimir Joseph Stephan Orlovsky, Mar 25 2011*)

From Elmo R. Oliveira, Feb 12 2024: (Start)
a(n) = 26 for n >= 1.
G.f.: 13*(1+x)/(1-x).
E.g.f.: 26*exp(x) - 13.
a(n) = 13*A040000(n). (End)

A041313 Denominators of continued fraction convergents to sqrt(170).

Original entry on oeis.org

1, 26, 677, 17628, 459005, 11951758, 311204713, 8103274296, 210996336409, 5494008020930, 143055204880589, 3724929334916244, 96991217912702933, 2525496595065192502, 65759902689607707985, 1712282966524865600112, 44585117032336113310897

Offset: 0

N. J. A. Sloane

From Michael A. Allen, May 04 2023: (Start)
Also called the 26-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 26 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 (26,1).

Cf. A041312, A040156.

Row n=26 of A073133, A172236 and A352361 and column k=26 of A157103.

Denominator[Convergents[Sqrt[170], 30]] (* Vincenzo Librandi, Dec 15 2013 *)
LinearRecurrence[{26,1},{1,26},20] (* Harvey P. Dale, Jul 26 2017 *)

a(n) = F(n, 26), the n-th Fibonacci polynomial evaluated at x=26. - T. D. Noe, Jan 19 2006
From Philippe Deléham, Nov 21 2008: (Start)
a(n) = 26*a(n-1) + a(n-2) for n > 1, a(0)=1, a(1)=26.
G.f.: 1/(1-26*x-x^2). (End)

An additional term from Colin Barker, Nov 15 2013

cp's OEIS Frontend

A040156 Continued fraction for sqrt(170).

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