A088209 Numerators of convergents of the continued fraction with the n+1 partial quotients: [1;1,1,...(n 1's)...,1,n+1], starting with [1], [1;2], [1;1,3], [1;1,1,4], ...

1, 3, 7, 14, 28, 53, 99, 181, 327, 584, 1034, 1817, 3173, 5511, 9527, 16402, 28136, 48109, 82023, 139481, 236631, 400588, 676822, 1141489, 1921993, 3231243, 5424679, 9095126, 15230452, 25475429, 42566379, 71052157, 118489383

Offset

0,2

Comments

Denominators form the Les Marvin sequence: A007502(n+1).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Formula

G.f.: (1+x-x^3)/(1-x-x^2)^2. [Corrected by Georg Fischer, Aug 16 2021]

a(n) = Fibonacci(n) + (n+1)*Fibonacci(n+1). - Paul Barry, Apr 20 2004

a(n) = a(n-1) + a(n-2) + Lucas(n). - Yuchun Ji, Apr 23 2023

Example

a(3)/A007502(4) = [1;1,1,4] = 14/9.

Mathematica

f[n_] := Numerator@  FromContinuedFraction@ Join[ Table[1, {n}], {n + 1}]; Array[f, 30, 0] (* Robert G. Wilson v, Mar 04 2012 *)
CoefficientList[Series[(1+x-x^3)/(-1+x+x^2)^2, {x, 0, 40}], x] (* or *) LinearRecurrence[{2, 1, -2, -1}, {1, 3, 7, 14}, 40] (* Harvey P. Dale, Jul 13 2021 *)

Prog

(Haskell)
a088209 n = a088209_list !! n
a088209_list = zipWith (+) a000045_list $ tail a045925_list
-- Reinhard Zumkeller, Oct 01 2012, Mar 04 2012
(Julia) # The function 'fibrec' is defined in A354044.
function A088209(n)
    n == 0 && return BigInt(1)
    a, b = fibrec(n)
    a + (n + 1)*b
end
println([A088209(n) for n in 0:32]) # Peter Luschny, May 18 2022

Crossrefs

a(n) = A109754(n, n+2) = A101220(n, 0, n+2).

Cf. A007502 (the denominators), A000045, A045925.

Sequence in context: A285447 A339655 A140741 * A089074 A125176 A153234

Adjacent sequences: A088206 A088207 A088208 * A088210 A088211 A088212

Keyword

frac,nonn

Author

Paul D. Hanna, Sep 23 2003

Status

approved