On the Domino Ring

Consider a set of \( \frac{ N (N+1) } { 2 } \) dominos which have faces of values 1 to N.
A ring of dominos is a closed loop of them, in which any 2 touching dominos display the same value.
Let \(F(N) \) be the minimum numbers of rings that are needed to use up all of these dominos.

Find the last three digits of \(\displaystyle \sum _{n = 1} ^{2016} F(n) \).


  • As an explicit example, \( F(3) = 1 \) because we have the ring listed in the above image.
  • For the purposes of this question, assume a single domino to be a ring. So \( F(2) = 3\) (see image below) because there are three rings.


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