9/26/2015
Not Sequence Numbers – But Kind of Like
Sequence Numbers
This does not fit the definition of a
sequence number. However, it does have
an interesting “sequence numberish” result, and may help someone else find a
sequence number for the Lucas Sequence.
The Lucas Sequence
The Lucas Sequence
1999999/999998999999 =
(Note: This fraction has been adapted
from mathematical work I found on the website: http://www.asahinet.or.jp/~kc2hmsm/mathland/math05/repeat05.htm I believe the name of the mathematician who
produced this website is Hisanori Mishima.
0.
000002 000001 000003 000004 000007 000011 000018 000029 000047 000076 000123 000199 000322 000521 000843 001364 002207 003571 005778 009349 015127 024476 039603 064079 103682 167761 271443 439204 …
Terms are written in six digit strings.
Terms are accurate up to 439204, which is
the 27^{th} nonzero term.
Compare with OEIS sequence A000032.


The fraction we started with above can be
rewritten as the sum of three unit fractions:
1999999/999998999999 =
1/500000 + 1/999997000005 + 1/99999600000699999799999500000
If you take each one of these unit
fractions and convert them into decimals there sum shows the Lucas
Sequence. (In the final sum, the last
digit is rounded up.
0.000002
0.000000000001000003000003999996999970999927999929… 0.000000000000000000000000000010000040000090000099… 0.000002000001000003000004000007000011000018000029…
You might also have noticed that the
denominators of the second and third fractions look like the sequence numbers
for two different Fibonacci like sequences.
999,997,000,005 is the sequence number
for a Fibonacci like sequence that is defined as: a(0) = 0, a(1) = 1, and
when n>1 then a(n) = a(n1) – 5*a(n2).
I would call this the 1, 5 Fibonacci sequence.
99,999,600,000,699,999,799,999,500,000
looks like a sequence number for a Fibonacci like sequence, which was then
multiplied by 100,000. The Fibonacci
like sequence is defined as: a(0) = a(1) = a(2) = 0, a(3) = 1, and when
n>3 then a(n) = 4*a(n1) – 7*a(n2) + 2*a(n3) + 4*a(n4). I would call this the 4, 7, 2, 4
Tetranacci Sequence.
Both of these sequences produce terms
that are negative. Negative terms
manifest themselves in the decimal expansion in a strange way. That is why the decimal expansions shown
above have the multiple occurrences of strings of nines where you would
expect to see strings of zeros.
This table show the results of each of
the three unit fractions used to represent the Lucas Sequence. And the fourth column shows the terms of
the Lucas Sequence.
The mathematics used to combine these
into a decimal expansion that shows the terms of the Lucas Sequence (written
in six digit strings) can be a bit confusing when you consider that some of
the terms in the table above contain more than six digits and some of the
terms are negative. It's a good thing that the math knows what it is doing!
Still, it is amazing that the Lucas
Sequence, which apparently cannot be expressed as one unit fraction, can be
expressed as the sum of three unit fractions.

(Note: 1999999/999998999999 This fraction has been adapted from
mathematical work I found on the website: http://www.asahinet.or.jp/~kc2hmsm/mathland/math05/repeat05.htm
I believe the name of the mathematician
who produced this website is Hisanori Mishima.
In mathematics we cannot credit all of the mathematicians who developed
the mathematics that we use to produce new findings. After all, who would I credit for first
finding out that one plus three equals four.
But when I started this work, and needed some guidance and a hint as to
which direction to proceed, I found Hisanori Mishima’s work very helpful.)
David
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