A097691 Denominators of the continued fraction n-1/(n-1/...) [n times].

1, 2, 8, 56, 551, 6930, 105937, 1905632, 39424240, 922080050, 24057287759, 692686638072, 21817946138353, 746243766783074, 27543862067299424, 1091228270370045824, 46187969968474139807, 2080128468827570457762, 99318726126650358502921, 5011361251329169946919800

Offset: 1

Author

T. D. Noe, Aug 19 2004

Keywords

• easy
• frac
• nonn

Comments

The (n-1)-th term of the Lucas sequence U(n,1). The numerator is the n-th term. Adjacent terms of the sequence U(n,1) are relatively prime.

Examples

			a(4) = 56 because 4-1/(4-1/(4-1/4)) = 209/56.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..387
• Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024. See pp. 8, 22.
• Pascual Jara and Miguel L. Rodríguez, Solving quadratic congruences, Arhimede Math. J. (2020) Vol. 7, No. 2, 105-120.
• Eric Weisstein's World of Mathematics, Lucas Sequence

Crossrefs

Cf. A084844, A084845, A097690 (numerators).

Programs

Mathematica

Table[s=n; Do[s=n-1/s, {n-1}]; Denominator[s], {n, 20}]
Table[Abs[Fibonacci[n, I n]], {n, 20}] (* Vladimir Reshetnikov, Oct 16 2018 *)

Sage

[lucas_number1(n,n,1) for n in range(1,19)] # Zerinvary Lajos, Jul 16 2008

Formula

a(n) = ChebyshevU(n-1,n/2). - Gary Detlefs, Oct 15 2011

a(n) = abs((2^(-n) * (sqrt(4 - n^2) + i*n)^n - 2^n*(-sqrt(4 - n^2) - i*n)^(-n))/sqrt(4 - n^2)), where i is the imaginary unit, for n > 2. - Daniel Suteu, May 31 2017

a(n) ~ n^(n-1). - Vaclav Kotesovec, Jun 03 2017