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December 8, 2016
Posted by **Robo123** on Saturday, December 11, 2010 at 1:06pm.

A1= 2

A2=5

An=An-1 + An-2, n≥3

- Math (sequence) -
**MathMate**, Saturday, December 11, 2010 at 4:02pmThis 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=An-1+An-2

Substitute An-1=An-2+An-3, we get

An=2An-2+An-3

Similarly, and continuing,

An=3An-3+2An-4

An=5An-4+3An-5

An=8An-5+5An-6

An=11An-6+8An-5

....

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=Fn-2*A1+Fn-1*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.