A084845 Numerators of the continued fraction n+1/(n+1/...) [n times].

1, 5, 33, 305, 3640, 53353, 927843, 18674305, 426938895, 10928351501, 309601751184, 9616792908241, 324971855514293, 11868363584907985, 465823816409224245, 19553538801258341377, 874091571490181406680

Offset: 1

Author

Hollie L. Buchanan II, Jun 08 2003

Keywords

• 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. - T. D. Noe, Aug 19 2004

Examples

			a(4) = 305 since 4+1/(4+1/(4+1/4)) = 305/72.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..386
• Eric Weisstein's World of Mathematics, Lucas Sequence

Crossrefs

Cf. A084844 (denominators).

Cf. A097690, A097691, A117715.

Programs

Maple

A084845 := proc(n)
    fibonacci(n+1,n) ;
end proc:
seq(A084845(n),n=1..20) ; # Zerinvary Lajos, Dec 01 2006

Mathematica

myList[n_] := Module[{ex = {n}}, Do[ex = {ex, n}, {n - 1}]; Flatten[ex]] Table[Numerator[FromContinuedFraction[myList[n]]], {n, 1, 20}]
Table[s=n; Do[s=n+1/s, {n-1}]; Numerator[s], {n, 20}] (* T. D. Noe, Aug 19 2004 *)

PARI

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

Python

from sympy import fibonacci
def a117715(n, m): return 0 if n==0 else fibonacci(n, m)
def a(n): return a117715(n + 1, n)
print([a(n) for n in range(1, 31)]) # Indranil Ghosh, Aug 12 2017

Formula

a(n) = Sum_{k=0..floor(n/2)}* binomial(n-k, k)*n^(n-2k). - Michel Lagneau

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

a(n) = (s^(n+1) - (-s)^(-n-1))/(2*s - n), where s = (n + sqrt(n^2 + 4))/2. - Vladimir Reshetnikov, May 07 2016

a(n) = A117715(n+1,n). - Alois P. Heinz, Aug 12 2017