This formula has 2 parts, first the numerator is the root of (x^3-x^2-x-1) no surprise here, but the denominator was obtained using LLL (Pari-Gp) algorithm. The thing is, if you try to get a closed formula by doing the Z-transform or anything classical, it won't work very well since the actual symbolic expression will be huge and won't simplify.
The numerical values of Tribonacci numbers are c**n essentially and the c here is one of the roots of (x^3-x^2-x-1), then there is another constant c2. So the exact formula is c**n/c2.
Another way of doing 'exact formulas' are given by using [ ] function the n'th term of the series expansion of 1/(1+x+x**2) is
1 - 2 floor(1/3 n + 2/3) + floor(1/3 n + 1/3) + floor(1/3 n)
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The twin primes constant.
0.660161815846869573927812110014555778432623
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The Varga constant, also known to be the 1/(one-ninth constant).
9.2890254919208189187554494359517450610317