## Other Sequences in the Fibonacci Sequence

Arithmetic, algebra, number theory, sequence and series, analysis, ...
elendiastarman
Posts: 410
Joined: Sat Dec 22, 2007 8:15 pm

### Other Sequences in the Fibonacci Sequence

I was curious about the relationship that every other term has to each other in the Fibonacci sequence, and so I worked out the relation, and this is what I found.

1, 2, 5, 13, 34, 89, ... follows the sequence Fn+1 = 3Fn - Fn-1.
1, 3, 8, 21, 55, 144, ... follows the same sequence as well.

The astonishing conclusion is that the Fibonacci sequence, obtained by simply adding together the two previous terms, is also a composition of two like sequences that differ only in their second term! Want some
3.14159265358979323846264338327950288419716939937510
58209749445923078164062862089986280348253421170679...? daniel.is.fischer
Posts: 2400
Joined: Sun Sep 02, 2007 11:15 pm
Location: Bremen, Germany

### Re: Other Sequences in the Fibonacci Sequence

From Fn+1 = Fn + Fn-1, you can deduce
Fn+2 = Fn+1 + Fn
= (Fn + Fn-1) + Fn
= 2*Fn + Fn-1
= 2*Fn + (Fn - Fn-2)
= 3*Fn - Fn-2.

Now find the general relation of the sequence (Fn, Fn+k, Fn+2*k, ...) for arbitrary k.
Il faut respecter la montagne -- c'est pourquoi les gypa&egrave;tes sont l&agrave;.

uws8505
Posts: 58
Joined: Tue Sep 30, 2008 3:13 pm
Location: South Korea

### Re: Other Sequences in the Fibonacci Sequence

Fn+3 = Fn+2 + Fn+1
= 2 Fn+1 + Fn
= 3 Fn + 2 Fn-1
= 3 Fn + Fn-1 + Fn-2 + Fn-3
= 4 Fn + Fn-3

Fn+4 = Fn+3 + Fn+2
= 2 Fn+2 + Fn+1
= 3 Fn+1 + 2 Fn
= 5 Fn + 3 Fn-1
= 5 Fn + 3 Fn-2 + 3 Fn-3
= 5 Fn + 6 Fn-3 + 3 Fn-4
= 5 Fn + 2 Fn-1 - 2 Fn-2 + 4 Fn-3 + 3 Fn-4
= 7 Fn - 4 Fn-2 + 4 Fn-3 + 3 Fn-4
= 7 Fn - Fn-4
Math and Programming are complements

uws8505
Posts: 58
Joined: Tue Sep 30, 2008 3:13 pm
Location: South Korea

### Re: Other Sequences in the Fibonacci Sequence

General formula:
Fn+k = Lk Fn + (-1)k+1 Fn-k

where Ln is Lucas number defined by L1 = 1, L2 = 3, Ln+2 = Ln+1 + Ln

Proof:
Expand
Math and Programming are complements

Metalith
Posts: 6
Joined: Wed Nov 24, 2010 5:39 am

### Re: Other Sequences in the Fibonacci Sequence

I was studying the fibonacci sequence the other day trying a pattern for finding every nth term in the fibonacci sequence and I discovered this sequence in it.

Fn=(2x-1 * Fn-1) + Fn-2

Where x is every xth term and ninitial is 0. I was googling and couldnt find this anywhere, could someone explain what this is. So if you wanted every term, you would end up with the regular equation as x would = 1

EDIT: For clarification n is not referring to the original fibonacci sequence but the sequence that shows every xth term. Oh and sorry, didnt realize the last post was from last year
Working my way up there. Susanne
Posts: 32
Joined: Sun Nov 08, 2009 7:39 am

### Re: Other Sequences in the Fibonacci Sequence

I tested the formula Fx*n = 2x-1 * Fx*(n-1) + Fx*(n-2) for the x-values of 1, 2, 3, 4 and 5. It matched for x = 1 and for x = 3.

F3*n = 8 * F3*n-5 + 5 * F3*n-6
F3*n = 8 * F3*n-5 + 4 * F3*n-6 + F3*n-6
F3*n = 4 * F3*n-4 + 4 * F3*n-5 + F3*n-6
F3*n = 4 * F3*n-3 + F3*n-6
F3*n = 22 * F3*(n-1) + F3*(n-2)

F5*n = 55 * F5*n-9 + 34 * F5*n-10
F5*n = 55 * F5*n-9 + 33 * F5*n-10 + F5*n-10
F5*n = 33 * F5*n-8 + 22 * F5*n- 9 + F5*n-10
F5*n = 22 * F5*n-7 + 11 * F5*n- 8 + F5*n-10
F5*n = 11 * F5*n-6 + 11 * F5*n- 7 + F5*n-10
F5*n = 11 * F5*n-5 + F5*n-10
F5*n = 11 * F5*(n-1) + F5(n-2)

11 [ne] 24, so x = 5 is an example that the formula does not always match. Metalith
Posts: 6
Joined: Wed Nov 24, 2010 5:39 am

### Re: Other Sequences in the Fibonacci Sequence

Woops, My math mustve been off when I checked other values.
Terribly sorry.
Well for the problem i was doing I only needed to work for 3 so I checked other values and thought they were. Again very sorry
Working my way up there. Susanne
Posts: 32
Joined: Sun Nov 08, 2009 7:39 am

### Re: Other Sequences in the Fibonacci Sequence

There is no need to be sorry. After all the pattern with x = 3 is very useful for the problem you solved. For x = 5 there is an analogical pattern and it is likely that other x-values with such patterns can be found. 