Math (sequence)
posted by Robo123 on .
Find the first four terms of each sequence.
A1= 2
A2=5
An=An1 + An2, n≥3

This is a recursive relationship very similar to the Fibonacci numbers, except that A1=2 and A2=5, whereas F1=1 and F2=1 for Fibonacci numbers.
We start with
An=An1+An2
Substitute An1=An2+An3, we get
An=2An2+An3
Similarly, and continuing,
An=3An3+2An4
An=5An4+3An5
An=8An5+5An6
An=11An6+8An5
....
Replacing the coefficients by Fibonacci numbers, where
F0=0, F1=1, F2=1, F3=2, F4=3...
and continuing the substitution until A2 and A1, we would get:
An=Fn2*A1+Fn1*A2
Thus
A4=F2*A1+F3*A2
=1*2+2*5
=12
A8
=F6*2+F7*5
=8*2+13*5
=16+65
=81
Work out A8 manually as a check.