Question

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Recurrence Relations

How do I find the n:th number in the tribonacci series?
I need and algorithm fast enough for n up to 10^15.

Tribonacci numbers are defined as a(n) = a(n-1) + a(n-2) + a(n-3) with a(0)=a(1)=0, a(2)=1.

Answer 1

For any sequence with a linear recurrence, the matrix exponentiation algorithm works.

If e.g. the sequence has the recurrence

a[k] = x*a[k-1] + y*a[k-2] + z*a[k-3]

for k >= 3 and initial values a[0], a[1], a[2], you obtain the triple (a[n+2], a[n+1], a[n]) by multiplying

|x y z|^n  |a[2]|
|1 0 0|  * |a[1]|
|0 1 0|    |a[0]|

The matrix can be raised to the n^th power using exponentiation by repeated squaring in O(log n) steps.