In a previous post (four years ago), I explained the Fibonacci Series - where it comes from, how it originates, and how it can appear in nature. I also talked about the golden ratio, which is the ratio between subsequent terms in the Fibonacci Series.

I've been doing some further reading and research on the Fibonacci Series, and have been doing some of my own calculations and investigations.

Firstly: what happens if we extend the series, so that instead of just summing the two previous terms, we sum the three previous terms, or the four or five previous?

This has been done before (I wasn't too surprised), and these are known as the following:

Fibonacci - 2 terms

Tribonacci - 3 terms

Tetranacci (or quadranacci) - 4 terms

Quintanacci (or pentanacci) - 5 terms

Hexanacci - 6 terms

I struggled to find names for the higher-number terms, so I'm going to submit my own.

Heptanacci - 7 terms

Octanacci - 8 terms

Nonancci - 9 terms

Decanacci - 10 terms

I stopped at 10, as I found that the data I'd accumulated was enough to draw some interesting conclusions from. Here are the first few terms of each of the series:

Fib: 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377

Trib: 0,0,1,1,2,4,7,13,24,44,81,149,274,504,927

I've been doing some further reading and research on the Fibonacci Series, and have been doing some of my own calculations and investigations.

Firstly: what happens if we extend the series, so that instead of just summing the two previous terms, we sum the three previous terms, or the four or five previous?

This has been done before (I wasn't too surprised), and these are known as the following:

Fibonacci - 2 terms

Tribonacci - 3 terms

Tetranacci (or quadranacci) - 4 terms

Quintanacci (or pentanacci) - 5 terms

Hexanacci - 6 terms

I struggled to find names for the higher-number terms, so I'm going to submit my own.

Heptanacci - 7 terms

Octanacci - 8 terms

Nonancci - 9 terms

Decanacci - 10 terms

I stopped at 10, as I found that the data I'd accumulated was enough to draw some interesting conclusions from. Here are the first few terms of each of the series:

Fib: 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377

Trib: 0,0,1,1,2,4,7,13,24,44,81,149,274,504,927

Tetra: 0,0,0,1,1,2,4,8,15,29,56,108,208,401,773

Quint: 0,0,0,0,1,1,2,4,8,16,31,61,120,236,464

Hex: ...0,0,1,1,2,4,8,16,32,63,125,248

Hept: ...0,0,1,1,2,4,8,16,32,64,127

Oct: ...0,1,1,2,4,8,16,32,64,128,255,509,1016,2028,4048,8080,16128

Non: ...0,1,1,2,4,8,16,32,64,128,256,511,1021,2040,4076,8144

Dec: ..,0,1,1,2,4,8,16,32,64,128,256,512,1023,2045,4088

Taking this raw data, I then started to look at the ratios between subsequent terms. We've seen previously that the Fibonacci series has the golden ratio 1.61803... or (1+ sqrt(5))/2 but what about the other series?

Fib 1.61803

Trib 1.83929

Tetra 1.92756

Quint 1.965948

Hex 1.983583

Hept 1.991964

Oct 1.996031

Non 1.998029

Dec 1.999019

I haven't identified the expressions for each of these but have found from research that the tetranacci constant satisfies x + x

Plotting the number of terms being summed (or the n-number for the series) against the ratio gives this graph.

The question is: will the ratio ever reach 2? It looks like the line will head towards 2 as an asymptote, but will it reach 2 if N increases?

The answer is no, and my proof is as follows:

Take the N=10 series as an example:

0,1,

We can see after the initial 0 and 1, that the first few terms are exactly double the previous one. This is because each term is the sum of

However, this doubling of terms eventually ends: when the sum no longer includes

1+1+2+4+8+16+32+64+128+256 = 512

1+2+4+8+16+32+64+128+256+512 = 102

At this point, the ratio falls below 2 (1023/512 = 1.998. This fall will occur for any and all series which follow the "sum of previous terms" pattern; as N increases, it just takes more terms, and the final ratio will get closer to 2, but will remain below it. As an aside, Wikipedia states that the ratio for an n-nacci series tends to the solution, x, of the equation (no proof given, although my data confirms it).

Next: I will look at what happens when we sum the previous term and

Fib 1.61803

Trib 1.83929

Tetra 1.92756

Quint 1.965948

Hex 1.983583

Hept 1.991964

Oct 1.996031

Non 1.998029

Dec 1.999019

I haven't identified the expressions for each of these but have found from research that the tetranacci constant satisfies x + x

^{-4}= 2.Plotting the number of terms being summed (or the n-number for the series) against the ratio gives this graph.

The answer is no, and my proof is as follows:

Take the N=10 series as an example:

0,1,

**1,2,4,8,16,32,64,128,256,512**,1023,2045,4088We can see after the initial 0 and 1, that the first few terms are exactly double the previous one. This is because each term is the sum of

**all**the previous non-zero terms, including both of the 1s. 1+1 = 2, 1+1+2 = 4, 1+1+2+4=8 etc. In this case, the ratio of each term to its previous term is 2, each term is exactly double the previous one.However, this doubling of terms eventually ends: when the sum no longer includes

*all*the previous terms, that is to say, when the sum no longer includes both of the 1s and all subsequent terms, then the ratio falls below 2. In the N=10 example above, the ratio falls below 2 when we reach 1023.1+1+2+4+8+16+32+64+128+256 = 512

1+2+4+8+16+32+64+128+256+512 = 102

**3**At this point, the ratio falls below 2 (1023/512 = 1.998. This fall will occur for any and all series which follow the "sum of previous terms" pattern; as N increases, it just takes more terms, and the final ratio will get closer to 2, but will remain below it. As an aside, Wikipedia states that the ratio for an n-nacci series tends to the solution, x, of the equation (no proof given, although my data confirms it).

Next: I will look at what happens when we sum the previous term and

*half*of the term before that...e.g. N = 1.5, N= 2.5, N=3.5 etc.
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