A249265 Sequence of distinct least nonnegative numbers such that the average of the first n terms is a Fibonacci number.

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.

Links

Table of n, a(n) for n=1..49.

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

Cf. A248982, A000045.

Sequence in context: A002251 A093545 A293845 * A259972 A193762 A268489

Adjacent sequences: A249262 A249263 A249264 * A249266 A249267 A249268

Keyword

nonn

Author

Derek Orr, Oct 23 2014

Status

approved