As we have seen the Fibonacci Sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...
 

The Lucas is 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, …...
 
 

If one takes any two successive terms in the Fibonacci and multiply the first terms by 3 and the second by 4, the sum of the result will always be a term in the corresponding Lucas series.

Thus if we take 2 and 3 in the Fibonacci, (3 * 2) +(4 * 3) = 18 (which is a term in the Lucas).

In fact by taking successive terms in the Fibonacci in this manner, we likewise generate successive terms in the Lucas.

An even more interesting result emerges when we apply the same procedure to the Lucas.

If we multiply the first term by 3 and the second by 4 and then divide the resulting sum by 5 (thus employing the best known Pythagorean triple 3.4 and 5)we generate successive terms in the corresponding Fibonacci Sequence.

4 and 7 for example are successive terms in the Lucas Series.

(3 * 4) + (4 * 7) = 40. When divided by 5 the result = 8 (which is a terms in the Fibonacci).

(Thus when we take 7 and 11 in the Lucas, we generate 13 in the Fibonacci!)
 
 

Again looking at the Fibonacci 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

The first set of triples is - as we have seen - 3, 4 and 5.
The next set (generated from 1, 2, 3, 5) is 5, 12 and 13.

The Lucas is 1, 3, 4, 7, 11, 18, 29, 47, 76,...

Now when we take the second two terms in the Lucas and combine them with the
first set of triples (first multiplying each terms by 3 and 4 respectively
before adding) we get
 

3 * 3 + (4 * 4) = 25. Then dividing by 5 we get 5.

Now when me take the next set of terms (4 and 7) and use the next set of
Pythagorean triples (5, 12 and 13) we
get 5 * 4 + (12 * 7) = 104. Then dividing by 13 we get 8 (which is the next
term in the corresponding Fibonacci sequence).

This pattern continues throughout.

The next set of triples (from 2, 3, 5 and 8) is 16, 30 and 34.

Taking the next terms in the Lucas 7 and 11 and applying the same procedure
we get

16 * 7 + (30 * 11) = 442. Then when we divide this sum by 34 we get 13
(which is the next term in the Fibonacci).

Just to illustrate with one more!

The next set of triples (derived from (3, 5, 8 and 13) is 39, 80 and 89.

The corresponding next successive terms in the Lucas are 11 and 18

Thus multiplying the first term by 39 and the second by 80 and adding we get

39 * 11 = (80 * 18) = 1869. Then dividing by 89 we get 21 (i.e. the next
term in the Fibonacci!)

Very interesting indeed!
 

The fascinating results do not end here!

If we go back to the second set of Pythagorean triples (5, 12 and 13) and
now apply it to all terms

we get initially
5 * 1 + (12 * 3) = 41. When we divide by 13 we get 3 ( + 2)

with the next terms (3 and 4) we get

5 * 3 + (12 * 4) = 63. When we divide by 13 we get 5 ( - 2)

As we have seen in the next case ( 4 and 7) we get 8

With the next terms (7 and 11) we get

5 * 7 + (12 * 11) = 167. When we divide by 13 we get 13 (- 2)

With the next terms (11 and 18) we get

5 * 11 + (12 * 18) = 271. When we divide by 13 we get 21 (- 2)

With the next terms (18 and 29) we get

5 * 18 + (12 * 29) = 438. When we divide by 13 we get 34 ( - 4)

With the next terms (29 and 47) we get

5 * 29 + (12 * 47) = 709. When we divide by 13 we get 55 (- 6)

Taking just one more (47 and 76) to illustrate we get

5 * 47 + (12 * 76) = 1147. When we divide by 13 we get 89 (- 10)

See the pattern!

Once again we generate successive terms in the Fibonacci sequence (with a
remainder).

However there is a fascinating pattern to the remainders. Once we get past
the term without a remainder (i.e. 8) the terms in the remainder themselves
follow a definite Fibonacci sequence. This can be seen by dividing by
- 2 so that we get 1, 1, 2, 3, 5, ....