how do you find 29th number in the showing sequence?

[Arithmetica]

Given this sequence :
1,1,2,4,7,13,...
How do you find 29th number in this sequence?

[rrabbit]

Each number is the sum of the previous three, with the first three given as 1, 1, 2. I don't have an explicit formula for the nth term, and it's really tedious to calculate, but you can use a spreadsheet to generate the sequence very easily.

The 29th term turns out to be 15,902,591.

[infotreasure]

On observation, we find that:

t(n) = 1 if n=1, 2,
= t(n-1) + t(n-2) if n=3
and
= t(n-1) + t(n-2) + t(n-3) if n>3

For example, we can test it for the 6th term t(6).

t(6) = t(5) + t(4) + t(3)
= 7 + 4 + 2 = 13, which is correct.

To find t(29), we first write it as

t(29) = t(28) + t (27) + t (26).

We can then reduce it by repeated substitutions for each term as a function of previous terms, using the functional forms as described above till we reach an expression which involves just t(1) & t(2), both of which can then be replaced with 1 to get the final answer as 15,902,591.

It may be useful to know that this series has a special name called Tribonacci Series, which is a development of the eponymous Fibonacci series (each term being a sum of the previous two terms), based on the observation of the proliferation pattern among rabbits by the Italian mathematician Fibonacci.

I hope this gives a comprehensive solution to this problem.

[cOPYcAT]

1-1
2-1
3-2
4-4
5-7
6-13
7-24
8-44
9-81
10-149
11-274
12-504
13-927
14-1,705
15-3,136
16-5,768
17-10,609
18-19,513
19-35,890
20-66,012
21-121,415
22-223,317
23-410,744
24-755,476
25-1,389,537
26-2,555,757
27-4,700,770
28-8,646,064
29-15,902,591

i did this in excel.