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A084845 Numerators of the continued fraction n+1/(n+1/...) [n times]. 9
1, 5, 33, 305, 3640, 53353, 927843, 18674305, 426938895, 10928351501, 309601751184, 9616792908241, 324971855514293, 11868363584907985, 465823816409224245, 19553538801258341377, 874091571490181406680 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..386

Eric Weisstein's World of Mathematics, Lucas Sequence

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

EXAMPLE

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

MAPLE

with(combinat, fibonacci):seq(fibonacci(i+1, i), i=1..17); # 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 *)

PROG

(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 map(a, range(1, 31)) # Indranil Ghosh, Aug 12 2017

CROSSREFS

Cf. A084844 (denominators).

Cf. A097690, A097691, A117715.

Sequence in context: A291846 A255927 A001828 * A198079 A098460 A087618

Adjacent sequences:  A084842 A084843 A084844 * A084846 A084847 A084848

KEYWORD

frac,nonn

AUTHOR

Hollie L. Buchanan II, Jun 08 2003

STATUS

approved

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Last modified April 7 12:45 EDT 2020. Contains 333305 sequences. (Running on oeis4.)