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A072523 Sum of remainders when n-th Fibonacci number is divided by all smaller Fibonacci numbers > 1. 2

%I

%S 0,0,0,1,3,5,10,15,28,46,79,120,207,330,540,867,1428,2293,3737,6009,

%T 9778,15808,25630,41370,67092,108483,175649,284022,459938,743945,

%U 1204113,1947712,3152386,5100237,8253262,13352465,21607324,34959920

%N Sum of remainders when n-th Fibonacci number is divided by all smaller Fibonacci numbers > 1.

%H Giovanni Resta, <a href="/A072523/b072523.txt">Table of n, a(n) for n = 1..1000</a>

%F Conjecture: lim n->inf F(n)/a(n) = sqrt(5)/2 where F(n) is the n-th Fibonacci number and therefore lim n->inf a(n)/a(n-1) = Phi (i.e. (sqrt(5)+1)/2 or lim n->inf F(n)/F(n-1)) - _Gerald McGarvey_, Jul 14 2004

%e The eighth Fibonacci number is 21; division by 2, 3, 5, 8,13 gives the remainders 1, 0, 1, 5, 8, so a(8) = 1 + 0 + 1+ 5 + 8 = 15.

%t Table[Total[Mod[Fibonacci[n],Fibonacci[Range[n-1]]]],{n,40}] (* _Harvey P. Dale_, Mar 18 2015 *)

%o (PARI) for(n=1,38,s=0; for(j=3,n-1,s=s+fibonacci(n)%fibonacci(j)); print1(s,","))

%K nonn

%O 1,5

%A _Amarnath Murthy_, Jul 31 2002

%E Edited and extended by _Klaus Brockhaus_, Aug 02 2002

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Last modified August 9 20:09 EDT 2020. Contains 336326 sequences. (Running on oeis4.)