SEQUENCE NUMBERS THAT PRODUCE FIBONACCI LIKE SEQUENCES: Part 2

9/24/2015

SEQUENCE NUMBERS THAT PRODUCE FIBONACCI LIKE SEQUENCES: Part 2

There was one issue with producing sequence numbers for Fibonacci like sequences that I did not get into in the last lesson.  That issue is what to do if you if you have to subtract a previous term, or subtract some number times a previous term.

The procedure is pretty much the same as before, except that you add the multiple that you are subtracting.  This may be more clear if I give you an example.

Suppose you wanted to create a sequence number for a 2, 3, -1, -2, -3 Pentanacci sequence written with six digit strings:

Pentanacci sequences need a total of 5 terms.

The first term will be 999999 -2, or 999997.

The second term will be 999999 – 2, or 999996.

The third term will be 999999 + 1, or 1000000.

The fourth term will be 999999 + 2, or 1000001.

The fifth and last term will be 999999 + 3 +1 or 1000003.

The next step, putting the terms together, gives a detail we have to address.  No three of our 5 parts are 7 digits long instead of 6 digits long.  Our extra digit will have to be “carried” and added to the next part. In this case all three of our extra (or seventh) digits is a 1.

The last (or fifth) part is 1000003.

When add the fourth part to the fifth part we get 1000002000003.

Then when we add the third part we get 1000001000002000003.

Now add the second part and we get 999997000001000002000003.

Finally we add the first part and we get 999,997,999,997,000,001,000,002,000,003 which should give us the appropriate digital expansion.  Since this sequence is listed in the OEIS I will need to either manually create the sequence, or set up a spreadsheet to produce the needed terms so that we can compare the results from the sequence number to a list of terms that we know are correct.

This is a list of the first 18 terms (the first 14 non-zero terms), which includes all terms up to and including 6 digit terms:

0
0
0
0
1
2
7
19
55
153
432
1,209
3,394
9,512
26,674
74,776
209,647
587,742

Now let’s check it out and see if it performs as advertised.

The 2, 3, -1, -2, -3 Pentanacci Sequence: 999,997,999,997,000,001,000,002,000,003 1/999997999997000001000002000003 = 0. 000000  000000  000000  000000  000001  000002  000007  000019  000055  000153  000432  001209  003394  009512  026674  074776  209647  587743  … Please notice that the terms are written in six digit strings like we wanted them.  Notice also that all of the terms are accurate up to the 13th non-zero term, but the 14th term is off by 1.  That is because the 15th term is a seven digit number, so the 7th digit “carries over” and adds to the 15th term.  We have seen that before so it is not unexpected. If we wanted to see if we could push this to 12 digit strings and have accuracy up to and including 12 digit terms – well, I’ll let you try it – OPPS, SORRY, I forgot I am a mathematician, a retired teacher, and retired professor – “this will be left as an exercise for the reader”.

Now I want to push on and show you the surprise ending I promised.  (All the best books have a surprise ending don’t they?)

Let’s look at the 2, -1 Fibonacci Sequence.  We can make this one so that it has terms listed in three digit strings.

We will need two parts:  999 and 999.

The first part will be 999 – 2, which is 997.

The second part will be 999 +1 +1 (since it is the last part), which is 1001.

When we put these two parts together we will have to “carry over” the fourth digit in the second part (a 1).  With this in mind when we put them together we get: 998,001.

998,001 is a sequence number we have seen before.  It was the first one I introduced to you.  It produced a counting sequence that counted from 000 to 997 accurately.

I was surprised to find out that counting was really a special version of the Fibonacci Sequence (or that this special version of the Fibonacci Sequence was really just a counting sequence).

SURPRISE!

David