A237268 a(1)=1; for n > 1, a(n) is the smallest F(m) > F(n) such that F(n) divides F(m), where F(k) denotes the k-th Fibonacci number.

1, 2, 8, 21, 55, 144, 377, 987, 2584, 6765, 17711, 46368, 121393, 317811, 832040, 2178309, 5702887, 14930352, 39088169, 102334155, 267914296, 701408733, 1836311903, 4807526976, 12586269025, 32951280099, 86267571272, 225851433717, 591286729879, 1548008755920

Offset

1,2

Comments

a(n) > Fibonacci(n) for n > 1.

{a(n)} = {F(2*n)} union {2} minus {0,3} where F(2*n) = A001906(n) = bisection of Fibonacci sequence.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (3,-1).

Formula

From Colin Barker, Jul 29 2014: (Start)

a(n) = 3*a(n-1) - a(n-2) for n>4.

G.f.: -x*(x^3 - 3*x^2 + x - 1) / (x^2 - 3*x + 1). (End)

a(n) = (((3 + sqrt(5))/2)^n - ((3 - sqrt(5))/2)^n)/sqrt(5) for n > 2. - Stefano Spezia, Apr 15 2022

Example

377 is the first Fibonacci number that is divisible by 13, the 7th Fibonacci number, so a(7) = 377.

Maple

A237268 := proc(n)
    coeftayl((1-x+3*x^2-x^3)/(x^2-3*x+1), x=0, n);
end proc:
seq(A237268(n), n=0..30); # Wesley Ivan Hurt, Aug 02 2014

Mathematica

Table[k=1; While[Mod[Fibonacci[k], Fibonacci[n]]!=0||Fibonacci[k]==Fibonacci[n], k++]; Fibonacci[k], {n, 1, 30}]
CoefficientList[Series[-(x^3 - 3 x^2 + x - 1)/(x^2 - 3 x + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 01 2014 *)

Prog

(PARI) Vec(-x*(x^3-3*x^2+x-1)/(x^2-3*x+1) + O(x^100)) \\ Colin Barker, Jul 29 2014

Crossrefs

Cf. A000045, A001906.

Sequence in context: A014409 A303721 A109782 * A216893 A264245 A123044

Adjacent sequences: A237265 A237266 A237267 * A237269 A237270 A237271

Keyword

nonn,easy

Author

Michel Lagneau, Feb 05 2014

Status

approved