The sequence {S(n)} is defined as follows:
S(1) = S(2) = S(3) = 1, and:
S(k+3) = S(k+2) + S(k+1) + S(k), for k=1,2,.....n

It is known that:
S(28) = 6090307, S(29)= 11201821, and:
S(30) = 20603361.

Determine the remainder when Σ_{k=1 to 28} S(k) is divided by 1000.

(In reply to spoiler solution; can it clue toward the intended method? by Charlie)

It's easily done in Excel, too, which led me to the same conclusion (and answer) as you.

WolframAlpha seems to be down at the moment, but I did notice that the difference between successive values is 1.8392867552142, which appears to be a well-known constant. I dare say there are related formulae, too.

Edited on February 2, 2016, 8:55 am

Posted by broll on 2016-02-02 08:54:27