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%I #14 Nov 01 2014 02:13:19
%S 1,3,2,6,13,5,26,8,53,93,21,177,34,328,55,599,89,1079,144,1924,233,
%T 3401,377,5969,610,10412,987,18067,1597,31207,2584,53688,4181,92037,
%U 6765,157281,10946,268016,17711,455551,28657,772519,46368,1307276,75025,2207953,121393,3722593,196418,6266068,317811
%N Sequence of distinct least positive numbers such that the average of the first n terms is a Fibonacci number.
%F Conjecture: a(n) = 2*a(n-2)+a(n-4)-2*a(n-6)-a(n-8) for n > 17. - _Colin Barker_, Oct 19 2014
%F Empirical g.f.: x*(x -1)*(40*x^15 +98*x^13 +4*x^11 +3*x^10 -80*x^9 +7*x^8 -2*x^6 -2*x^5 -12*x^4 -4*x^3 -4*x^2 -4*x -1) / (x^4 +x^2 -1)^2. - _Colin Barker_, Oct 19 2014
%F Conjecture: For n > 4, a(2*n+1) = A000045(n+3).
%e 1/1 = 1 is a Fibonacci number. So a(1) = 1.
%e (1+2)/2 is not a Fibonacci number. (1+3)/2 = 2 is a Fibonacci number. So a(2) = 3.
%e (1+3+2)/3 = 2 is a Fibonacci number. So a(3) = 2.
%o (PARI) v=[]; n=1; while(n<10^7, num=(vecsum(v)+n); if(num%(#v+1)==0&&vecsearch(vecsort(v), n)==0, for(i=1, n+2, if(fibonacci(i)>(num/(#v+1)), break); if(fibonacci(i)==(num/(#v+1)), print1(n, ", "); v=concat(v, n); n=1; break))); n++)
%K nonn
%O 1,2
%A _Derek Orr_, Oct 18 2014
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