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).
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:
(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 =
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?