A097690 Numerators of the continued fraction n-1/(n-1/...) [n times].

1, 3, 21, 209, 2640, 40391, 726103, 15003009, 350382231, 9127651499, 262424759520, 8254109243953, 281944946167261, 10393834843080975, 411313439034311505, 17391182043967249409, 782469083251377707328

Offset: 1

Author

T. D. Noe, Aug 19 2004

Keywords

• easy
• frac
• nonn

Comments

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

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..386
• Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024. See pp. 3, 8, 10, 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
• Wikipedia, Chebyshev polynomials.

Crossrefs

Cf. A084844, A084845, A097691 (denominators), A179943, A323118.

Programs

Mathematica

Table[s=n; Do[s=n-1/s, {n-1}]; Numerator[s], {n, 20}]
Table[DifferenceRoot[Function[{y, m}, {y[1 + m] == n*y[m] - y[m - 1], y[0] == 1, y[1] == n}]][n], {n, 1, 20}] (* Benedict W. J. Irwin, Nov 05 2016 *)

PARI

{a(n)=polcoeff(1/(1-n*x+x^2+x*O(x^n)), n)} \\ Paul D. Hanna, Dec 27 2012

PARI

a(n) = polchebyshev(n, 2, n/2); \\ Seiichi Manyama, Mar 03 2021

PARI

a(n) = sum(k=0, n, (n-2)^k*binomial(n+1+k, 2*k+1)); \\ Seiichi Manyama, Mar 03 2021

Sage

[lucas_number1(n,n-1,1) for n in range(19)] # Zerinvary Lajos, Jun 25 2008

Formula

a(n) = [x^n] 1/(1 - n*x + x^2). - Paul D. Hanna, Dec 27 2012

a(n) = y(n,n), where y(m+1,n) = n*y(m,n) - y(m-1,n) with y(0,n)=1, y(1,n)=n. - Benedict W. J. Irwin, Nov 05 2016

From Seiichi Manyama, Mar 03 2021: (Start)

a(n) = U(n,n/2) where U(n,x) is a Chebyshev polynomial of the second kind.

a(n) = Sum_{k=0..n} (n-2)^(n-k) * binomial(2*n+1-k,k) = Sum_{k=0..n} (n-2)^k * binomial(n+1+k,2*k+1). (End)