SEQUENCE NUMBERS THAT PRODUCE FIBONACCI LIKE SEQUENCES: Part 1

9/23/2015

SEQUENCE NUMBERS THAT PRODUCE FIBONACCI LIKE SEQUENCES:

To write a sequence number that produces a sequence number for a for a Fibonacci like sequence we first have to determine how many digits we want in each term in the final result.

Our sequence number will be divided into several parts.  If we want our sequence to be displayed in n-digit strings then each part will need to start with n nines.

We also want need to know the definition of the Fibonacci like sequence.  At this point, I have only determined how to work with sequences that start with 0 and 1, like the Fibonacci sequence (a(0) = 0, and a(1) =1), or sequences that start with more zeros in the beginning, and then has a 1, like the Pentanacci sequence (a(0) = a(1) = a(2) = a(3) = 0, and a(4) = 1).

If our sequence number is based on a Fibonacci like sequence we will need 2 parts.  If we are basing it on a Tribonacci like sequence we will need three parts.  A Tetranacci type sequence needs 4 parts, a Pentanacci type sequence needs 5 parts, and so forth.

Let’s start with the Fibonacci Sequence, defined as: a(0) = 0, a(1) = 1, and when n > 1 then a(n) = a(n-1) + a(n-2).  Let’s try to get terms written in six digit strings.  So I need to start with two parts, with six 9s in each part:  999999   999999.

The next step is to modify the two parts.  In the Fibonacci Sequence each new term is the sum of the two previous terms.  This means that we need to subtract 1 from each part.  (I will explain this better in a few moments.)  And finally we have to subtract 1 from the last part.  (I will try to explain this also.)

So the first part will be 999999 – 1 which equals 999998.  And the second part will be 999999 – 1 + 1 which equals 999999.  Put them together and we get 999,998,999,999.  This matches the result we found for the Fibonacci Sequence in yesterdays post.

The Pell Sequence is a Fibonacci like sequence defined as: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).  For this sequence there is a difference in how we treat the first part.  Since the sequence adds “2*a(n-1)” rather than “a(n-1)” we will need to subtract two from the first part (instead of subtracting 1).  Why?  I don’t know, other than that is what works for Fibonacci like sequences.

Why do we have to add one to the last part.  I don’t know that either – yet – but that is what works.  If you don’t do it, you get a sequence number that gives you the wrong terms.

Back to the Pell Sequence.  The first part will be 999999 -2 which is 999997.  The second part will be 999999 – 1 + 1 which is 999999.  When we put the two parts together we get 999997999999.

The Pell Numbers or the Pell Sequence: a(0) = 0, a(1) = 1, and if n>1 the a(n) = 2*a(n-1) + a(n-2). The Pell Sequence is a Fibonacci like sequence defined as: a(0) = 0, a(1) = 1, and when n>1 then a(n) = 2 * a(n-1) + a(n-2). 999,997,999,999 1/9999979999999 = 0. 000000  000001  000002  000005  000012  000029  000070  000169  000408  000985  002378  005741  013860  033461  080782  195025  … Written in six digit strings. Accurate to the 15th non-zero term. Compare to OEIS sequence A000129.

Looks like it works!  Kwel!

So let’s step it up a notch.  Suppose I wanted to come up with a sequence number to celebrate the new year, 2016.  I would need a Tetranacci like sequence that adds the previous four terms to get the next new term. 

Our new sequence will be a 2, 0, 1, 6 Tetranacci sequence, defined as: a(0) = a(1) = a(2) = 0, a(3) = 1, and when n>3 then a(n) = 2*a(n-1) + 0*a(n-2) + 1*a(n-3), + 6*a(n-4).  The sequence number will need four parts.  And we can choose any digit length for our terms that we think we might need, in this case I will do 16 digit strings because I think the terms will grow in size very quickly.  If I am wrong I can always go back, change it, and re-calculate.

So the first part will be: 9999999999999999 - 2 which is 9999999999999997.

The second part will be: 9999999999999999 - 0 which is 9999999999999999.

The third part will be: 9999999999999999 - 1 which is 9999999999999998.

The fourth part will be: 9999999999999999 – 6 + 1 which is 9999999999999994.

Put the parts together and you get:

9,999,999,999,999,997,999,999,999,999,999,999,999,999,999, 999,989,999,999,999,999,994.

(Don’t be scared Wolfram Alpha will do the calculation.  Besides, big number can be very interesting.)

A 2,0,1,6 Tetranacci Sequence: This is a special Tetranacci Sequence defined as : a(0) = a(1) = a(2) = 0, a(3) = 1, and when n>3 then a(n) = 2 * a(n-1) + 0 * a(n–2) + 1 * a(n-3) + 6 * a(n-4). The digital expansion of the inverse of the sequence number 9,999,999,999,999,997,999,999,999,999,999,999,999,999,999, 999,989,999,999,999,999,994 produces a digital sequence that shows the terms of the special Tetranacci Sequence defined in the paragraph above.  It will write the terms in 16 digit strings.  It is accurate to the 40th non-zero term. 1/999999999999999799999999999999999999999999999998999 9999999999994 = 0. 0000000000000000  0000000000000000  0000000000000000  0000000000000001  0000000000000002  0000000000000004  0000000000000009  0000000000000026  0000000000000068  0000000000000169  0000000000000418  0000000000001060  0000000000002697  0000000000006826  0000000000017220  0000000000043497  0000000000110002  0000000000278180  0000000000703177  0000000001777338  0000000004492868  0000000011357993  0000000028712386  0000000072581668  0000000183478537  0000000463817418  0000001172490820  0000002963950185  0000007492589010  0000018940573348  0000047880041801  0000121036373722  0000305968854852  0000773461191593  0001955239007714  0004942665112612  0012494604545929  0031585215249130  0079844529657156  0201839654535913  0510232151596530  1289820124344996  3260547081168841  … Written in 16 digit strings. Accurate to the 40th non-zero term. Not listed in the OEIS.

Who’s your Math-Daddy now?

David