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A249265
Sequence of distinct least nonnegative numbers such that the average of the first n terms is a Fibonacci number.
0
0, 2, 1, 5, 7, 3, 17, 29, 8, 58, 13, 109, 21, 203, 34, 370, 55, 667, 89, 1189, 144, 2102, 233, 3689, 377, 6435, 610, 11166, 987, 19287, 1597, 33181, 2584, 56882, 4181, 97205, 6765, 165643, 10946, 281546, 17711, 477443, 28657, 807941, 46368, 1364590, 75025, 2300689, 121393
OFFSET
1,2
COMMENTS
This is the same concept as A248982, but with a(1) = 0, the zeroth Fibonacci number.
FORMULA
Conjecture: a(n) = 2*a(n-2) + a(n-4) - 2*a(n-6) - a(n-8) for n > 15. - Colin Barker, Oct 24 2014
Empirical g.f.: x^2*(x -1)*(12*x^12 + 28*x^10 + 3*x^9 - 2*x^8 + 7*x^7 - 24*x^6 - 2*x^5 - 9*x^3 - 4*x^2 - 3*x - 2) / (x^4 + x^2 - 1)^2. - Colin Barker, Oct 24 2014
For n > 5, a(2*n-3) = A000045(n).
EXAMPLE
a(4) = 5 because neither 3 nor 4 works (the average of {0, 2, 1, 3} is 3/2, while the average of {0, 2, 1, 4} is 7/4) and 5 does work, as the average of {0, 2, 1, 5} is 2, which is a Fibonacci number.
a(5) = 7 because neither 3, 4 nor 6 works (the averages would be 11/5, 12/5, 14/5, respectively) and 7 does work, as the average of {0, 2, 1, 5, 7} is 3, which is a Fibonacci number.
PROG
(PARI) v=[]; n=0; while(n<10^7, num=(vecsum(v)+n); if(num%(#v+1)==0&&vecsearch(vecsort(v), n)==0, for(i=0, n+4, if(fibonacci(i)>(num/(#v+1)), break); if(fibonacci(i)==(num/(#v+1)), print1(n, ", "); v=concat(v, n); n=0; break))); n++)
CROSSREFS
Sequence in context: A002251 A093545 A293845 * A259972 A193762 A268489
KEYWORD
nonn
AUTHOR
Derek Orr, Oct 23 2014
STATUS
approved