An Introduction to
the Strange and Beautiful World of
Sequence Numbers
By David Brooks
Mathematics is not a
careful march down a wellcleared highway, but a journey into a strange
wilderness, where the explorers often get lost. Rigour should be a signal to the historian
that the maps have been made, and the real explorers have gone elsewhere.
 W. S. Anglin

A few years ago I saw a post on a website
that showed that the inverse of 998,001 produces a decimal expansion that
counts, using three digit strings, from 000 to 997 without error.
I immediately thought that this had to be a
hoax. I decided to work it out to prove
it was a hoax – after all some people put anything they want on the web whether
it is true or not.
My calculator cannot handle more than eight
digits. And I really did not want to
work it out by hand. The arithmetic is
simple enough, I just did not want to spend that much time on it (we all like
instant everything – personally I get impatient waiting on the microwave). So I took this problem to a wonderful website,
Wolfram Alpha: www.wolframalpha.com .
It did the calculation for me, and I
checked it out – digit by digit. I did
not like what I found out – the equation was right, and I was wrong.
But I did start wondering – I wondered if
there were other numbers that had similar properties as this one. Are there other integers that, when you turn
the number into a unit fraction and then convert it to a decimal, produce an
interesting sequence of numbers (like counting, or multiplying, or whatever).
So I looked on the web for answers – but the
internet did not have any answers to this question (stupid internet!). Could it be true that the example I found in
a blog post was the only example of this special property?
I contacted a few other mathematicians by
email to see if they new anything about this kind of a mathematics – most did
not – but a few gave me so hints on where I might find out more. But those websites only gave me more hints.
By this time I realized two things – I was
not going to find an answer out there – and if I wanted to know the answer I
was going to have to find it myself. And
I really wanted to know the answer!
Things were slow at first, and at second,
and at third … but I found a few more hints, and eventually found another number
with a similar properties. Then I found
a few more.
But I did not understand why some numbers worked
and others did not.
So I kept working on the problem, and as I
found more of these special numbers I started to see patterns, and I learned
how to fill in some blank spots in these patterns.
I decided this group of numbers needed a
special name – after all I could not keep calling them just “special
numbers. “Mr. B,s Superduperfantastic
Arithmetically Accurate and Mathematically Mystical Sequence Numbers”. This name is really kind of long and it did
not catch on. Now I just call them “Sequence
Numbers” because they produce interesting sequences.
I define sequence numbers as integers that
produce interesting sequences when you take their inverses and convert them
into decimal form. I prefer that the “interesting
sequence” is listed in the Online Encyclopedia of Integer Sequences” (www.oeis.org ), but if the sequence is an
obvious or well known sequence I can accept that too.
So what are some of these other numbers? Let me tell you about some of my new buddies:
9,899 produce the first few terms of the
well known Fibonacci sequence.
196,020 produce a list of the multiples of
five.
998,999 produces the terms of the Fibonacci sequence, using three digit strings. 1/998999 = 0. 000 001 002 003 005 008 013 021 ...
998,998 produces the terms of the
Jacobsthal sequence, using three digit strings.
1/998998 = 0. 000 001 001
003 005 011
021 043 085 171 …
76,922,923,077 counts by multiples of 13. (Imagine that!
How far can you count by multiples of 13?)
999,999,700,000,029,999,999 shows a list of
all of the triangular numbers with 6 digits or less, and even most of the triangular
numbers with seven digits.
999,999,999,997 lists all of the powers of
two (2^{0}, 2^{1}, 2^{2}, 2^{3}, etc.), up to
12 digits long, except for the largest 12 digit power of 2. And it writes them all in 12 digit strings.
999,999,998,999,999,998,999,999,998,999,999,998,999,999,998,999,
999,998,999,999,998,999,999,998,999,999,998,999,999,998,999,999,
998,999,999,998,999,999,999 shows the terms of the tridecanacci sequence
(defined as: : a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) =
a(9) = a(10) = a(11) = 0, a(12) = 1 and when n>12 then a(n) = a(n1) +
a(n–2) + a(n3) + a(n4) + a(n 5) + a(n6) + a(n7) + a(n8) + a(n9) +
a(n10) + a(n11) + a(n12) + a(n13).
999,999,999,876,543,210 this one shows the
123,456,789 times table (all of the multiples of 123,456,789 – up to, but not
including the last 24 digit multiple of 123,456,789.
There are a lot of these special numbers out there, and I know I have not found them all
I hope some of you don’t believe me. Well let me rephrase that – I hope that some
of you don’t believe me AND have the gumption to try to prove me wrong. If you do you will learn how to find these
numbers and even how to design some to do specific things.
If you find something you want to show me, or you want me to post for others, please email me at: mbiom dot edu@gmail dot com.
Check out my website at:
SequenceNumbers.blogspot.com.
I also have another math website at: MathematicalMysteryTour.blogspot.com
David
https://beta.wikiversity.org/wiki/規律數列
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