Eureka!
It is possible when dealing with some of
these number sequences that you may have terms in the sequence that are negative
terms. However in the digital sequence
produced by the Sequence Numbers do not show negative terms when you do the
calculation. Instead you will see terms
that begin with nines.
In the past I did not use sequences that
produce negative terms because I had not worked why the negative terms produces
such strange results when produced by a Sequence Number.
This evening, when I was trying to go to
sleep, this problem was rolling around in my head. Somehow it bumped something loose, and I
suddenly realized how and why the terms looked so strange in the decimal
expansions.
This is because negative terms have to
borrow 1 from the previous term (because it cannot show a negative term). Each 1 borrowed from the previous term is
equal to 1000000 in this term (if the terms are expressed using six digit
strings) which must be added to the negative number, and the 1 that is borrowed
from the previous term must be subtracted from the previous term. So each term may change because it had to
borrow from the previous term and/or the next term had to borrow 1 from it.
Example: Please take a look at the next
sequence and the table that follows.
The 1, -2, 1 Tribonacci Sequence
This sequence is defined as: a(0) = a(1)
= 0, a(2) = 1, and
when n> 2, then a(n) = a(n-1) – 2*a(n-2) + a(n-3). The table below this box shows a comparison of the results of the recursive definition above, with the results of the Sequence Number shown below.
A Sequence Number that produces this
sequence is:
999,999,000,001,999,999
1/999999000001999999 =
0.
000000 000000 000001 000000 999998 999998 000001 000003 999999 999992 999997 000011 000009 999984 999976 000016 000048 999992 999910 999974 000145 000107 999791 999721 000245 000594 999825 998880 999824 001888 001120 997168 996815 003598 007136 996755 986079 999705 024301 010970 962073 964433 …
Each term is written in six digit
strings.
Each term shown above is accurate if you
use the correct
“Sequence Number” math. If the term is negative and had to borrow 1 from the previous term you have to add 1000000 to this term (because 1 in the previous term equals 1000000 in this term). If the next term had to borrow 1 from this term, then you have to subtract the 1 that was borrowed. Please see the table below for an explanation.
This sequence is not listed in the OEIS
collection.
|
The 1,
-2, 1 Tribonacci Sequence
|
|||
Term
|
Using
the
Recursive Definition |
Using
the
Sequence Number |
Notes
|
a(0)
|
0
|
000000
|
This
term is a 0. It does not need to
borrow anything from the previous term, and the next term does not need to
borrow 1 from it – so it does not change.
|
a(1)
|
0
|
000000
|
This
term is a 0. It does not need to
borrow anything from the previous term, and the next term does not need to
borrow 1 from it – so it does not change.
|
a(2)
|
1
|
000001
|
This
term is a 1. It does not need to
borrow anything from the previous term, and the next term does not need to
borrow 1 from it – so it does not change.
|
a(3)
|
1
|
000000
|
This
term is a 1, but the next term had to borrow the 1 because it is negative.
000001 –
1 = 000000
|
a(4)
|
-1
|
999998
|
This
term is a -1. It had to borrow 1 from
the previous term and subtract 1 from it.
Then next term is also negative, so it borrowed one from this term.
1000000
– 1 – 1 = 999998
|
a(5)
|
-2
|
999998
|
This
term is a -2. It had to borrow 1 from
the previous term and then subtract 2 from it.
1000000
– 2 = 999998
|
a(6)
|
1
|
000001
|
This
term is a 1. It does not need to
borrow anything from the previous term, and the next term does not need to
borrow 1 from it – so it does not change.
|
a(7)
|
4
|
000003
|
This
term is a 4, but the next term had to borrow 1 from it.
000004 –
1 = 000003
|
a(8)
|
0
|
999999
|
This
term is a 0. The next term is
negative, it needs to borrow 1. In
order to have 1 for it to borrow, this term needs to borrow 1 from the
previous term.
1000000
– 1 = 999999
|
a(9)
|
-7
|
999992
|
This
term is a -7. It had to borrow 1 from
the previous term, and the next term had to borrow 1 from this term.
1000000
– 7 – 1 = 999992
|
a(10)
|
-3
|
999997
|
This
term is a -3. It had to borrow 1 from
the previous term.
1000000
– 3 = 999997
|
a(11)
|
11
|
000011
|
This
term is an 11. It does not need to
borrow anything from the previous term, and the next term does not need to
borrow 1 from it – so it does not change.
|
a(12)
|
10
|
000009
|
This
term is a 10. However, the next term
is negative and had to borrow 1 from it.
000010 –
1 = 000009
|
a(13)
|
-15
|
999984
|
This
term is a -15. It had to borrow 1 from
the previous term, and the next term had to borrow 1 from it.
1000000
- 15 – 1 = 999984
|
a(14)
|
-24
|
999976
|
This
term is a -24. It had to borrow 1 from
the previous term.
1000000
– 24 = 999976
|
a(15)
|
16
|
000016
|
This
term is a 16. It did not have to
borrow 1, or lend 1 to the next term – so it did not change.
|
I am celebrating this new insight the same way Archimedes did - by running around naked and yelling "Eureka!". (I think "Eureka" is the Greek word for "Wow, I think I found the answer!")
Please excuse me while I put my pajamas back on, and go back to bed. In the mean time please close your eyes.
David
