1.8392867552141611325518525646532866004241787460975922467787586394042032220819\ 664257384354194283070141419798268592409741641784507465074369438315458204995137\ 962496555396446136661215402779726781189410412116092232821559560718167121823659\ 866522733785378156969892521173957914132287210618789840852549569311453491349853\ 459576175035965221323814247272722417358187700069790551025490449657107425265477\ 228110065989375556363093330528262357538519719942991453008254663977472900587005\ 974481391931672825848839626332970700687236831127837750250557122275153259578946\ 560570686422283918659698294691356239220443192476147068811451726766712743964146\ 212571843342662340390218352494591033227231061513286997030808036302223324997105\ 243107472354231399744381826565607351940357874911762680524537079221110849710806\ 876410050156541475662235008885665949715821834184868714802901255436993480513679\ 165025853053878276666126224317766358200942985505387325991651787730184472388604\ 26222324857820792721049160181783725613203439814302274533997621231 / 19 1/2\1/3 4 |---- + 1/9 33 | + ----------------------- + 1/3 \ 27 / / 19 1/2\1/3 9 |---- + 1/9 33 | \ 27 / in fact the n'th Tribonacci number is given by this EXACT formula. --------------------------------------------------------------------- See : http://www.labri.u-bordeaux.fr/~loeb/book/92pl.html Comment calculer le nieme nombre de Tribonacci 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