Eureka!

11/29/2015 at 3:13 AM

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