cp's OEIS Frontend

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

%I A097690 #43 Feb 16 2025 08:32:54
%S A097690 1,3,21,209,2640,40391,726103,15003009,350382231,9127651499,
%T A097690 262424759520,8254109243953,281944946167261,10393834843080975,
%U A097690 411313439034311505,17391182043967249409,782469083251377707328
%N A097690 Numerators of the continued fraction n-1/(n-1/...) [n times].
%C A097690 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.
%H A097690 Alois P. Heinz, <a href="/A097690/b097690.txt">Table of n, a(n) for n = 1..386</a>
%H A097690 Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, <a href="https://arxiv.org/abs/2406.12941">Metered Parking Functions</a>, arXiv:2406.12941 [math.CO], 2024. See pp. 3, 8, 10, 22.
%H A097690 Pascual Jara and Miguel L. Rodríguez, <a href="https://amj-math.com/wp-content/uploads/2022/01/AMJ2020-vol7iss2.pdf#page=6">Solving quadratic congruences</a>, Arhimede Math. J. (2020) Vol. 7, No. 2, 105-120.
%H A097690 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasSequence.html">Lucas Sequence</a>
%H A097690 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials">Chebyshev polynomials</a>.
%F A097690 a(n) = [x^n] 1/(1 - n*x + x^2). - _Paul D. Hanna_, Dec 27 2012
%F A097690 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
%F A097690 From _Seiichi Manyama_, Mar 03 2021: (Start)
%F A097690 a(n) = U(n,n/2) where U(n,x) is a Chebyshev polynomial of the second kind.
%F A097690 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)
%e A097690 a(4) = 209 because 4-1/(4-1/(4-1/4)) = 209/56.
%t A097690 Table[s=n; Do[s=n-1/s, {n-1}]; Numerator[s], {n, 20}]
%t A097690 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 *)
%o A097690 (Sage) [lucas_number1(n,n-1,1) for n in range(19)] # _Zerinvary Lajos_, Jun 25 2008
%o A097690 (PARI) {a(n)=polcoeff(1/(1-n*x+x^2+x*O(x^n)), n)} \\ _Paul D. Hanna_, Dec 27 2012
%o A097690 (PARI) a(n) = polchebyshev(n, 2, n/2); \\ _Seiichi Manyama_, Mar 03 2021
%o A097690 (PARI) a(n) = sum(k=0, n, (n-2)^k*binomial(n+1+k, 2*k+1)); \\ _Seiichi Manyama_, Mar 03 2021
%Y A097690 Cf. A084844, A084845, A097691 (denominators), A179943, A323118.
%K A097690 easy,frac,nonn
%O A097690 1,2
%A A097690 _T. D. Noe_, Aug 19 2004