Other Sequences in the Fibonacci Sequence

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elendiastarman
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Other Sequences in the Fibonacci Sequence

Post by elendiastarman »

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! :shock:
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daniel.is.fischer
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Re: Other Sequences in the Fibonacci Sequence

Post by daniel.is.fischer »

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 gypaètes sont là.

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uws8505
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Re: Other Sequences in the Fibonacci Sequence

Post by uws8505 »

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

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uws8505
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Re: Other Sequences in the Fibonacci Sequence

Post by uws8505 »

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:
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Induction step only :)
Fn+k+1 = Fn+k + Fn+k-1 = Lk Fn + (-1)k+1 Fn-k + Lk-1 Fn + (-1)k Fn-k+1

= (Lk + Lk-1) Fn + (-1)k (Fn-k+1 - Fn-k) = Lk+1 Fn + (-1)k+2 Fn-k-1
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Metalith
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Re: Other Sequences in the Fibonacci Sequence

Post by Metalith »

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.
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Susanne
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Re: Other Sequences in the Fibonacci Sequence

Post by Susanne »

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.
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Metalith
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Re: Other Sequences in the Fibonacci Sequence

Post by Metalith »

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.
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Susanne
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Re: Other Sequences in the Fibonacci Sequence

Post by Susanne »

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.
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