A237268 - OEIS

%I #25 May 05 2022 03:26:29

%S 1,2,8,21,55,144,377,987,2584,6765,17711,46368,121393,317811,832040,

%T 2178309,5702887,14930352,39088169,102334155,267914296,701408733,

%U 1836311903,4807526976,12586269025,32951280099,86267571272,225851433717,591286729879,1548008755920

%N 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.

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

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

%H Vincenzo Librandi, <a href="/A237268/b237268.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1).

%F From _Colin Barker_, Jul 29 2014: (Start)

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

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

%F a(n) = (((3 + sqrt(5))/2)^n - ((3 - sqrt(5))/2)^n)/sqrt(5) for n > 2. - _Stefano Spezia_, Apr 15 2022

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

%p A237268 := proc(n)

%p     coeftayl((1-x+3*x^2-x^3)/(x^2-3*x+1), x=0, n);

%p end proc:

%p seq(A237268(n), n=0..30); # _Wesley Ivan Hurt_, Aug 02 2014

%t Table[k=1;While[Mod[Fibonacci[k],Fibonacci[n]]!=0||Fibonacci[k]==Fibonacci[n],k++];Fibonacci[k], {n,1,30}]

%t CoefficientList[Series[-(x^3 - 3 x^2 + x - 1)/(x^2 - 3 x + 1), {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 01 2014 *)

%o (PARI) Vec(-x*(x^3-3*x^2+x-1)/(x^2-3*x+1) + O(x^100)) \\ _Colin Barker_, Jul 29 2014

%Y Cf. A000045, A001906.

%K nonn,easy

%O 1,2

%A _Michel Lagneau_, Feb 05 2014