Resume of a conference given in 1993 (Universite Bordeaux I, LaBRI).
1/2 1/3 1/2 1/3 n 1/2 1/3
(1/3 (19 + 3 33 ) + 1/3 (19 - 3 33 ) + 1/3) (586 + 102 33 )
3 ---------------------------------------------------------------------------
1/2 2/3 1/2 1/3
(586 + 102 33 ) + 4 - 2 (586 + 102 33 )
To get the actual n'th Tribonacci number just round the result to the
nearest integer.
Here is the formula 'lprinted'...
3*(1/3*(19+3*33^(1/2))^(1/3)+1/3*(19-3*33^(1/2))^(1/3)+1/3)^n/((586+102*33^(1
/2))^(2/3)+4-2*(586+102*33^(1/2))^(1/3))*(586+102*33^(1/2))^(1/3);
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