1/9801 and even deeper examples

Jul 2018
7
0
UK
These reciprocals are nothing new, but are surprising and link into other areas of maths and many types of sequence. I've seen the terms generators and generating sequences used with these, but haven't seen more than the simplest examples elsewhere.

1/9801 = 00010203…
1/998001 = 000001002003…009010011012013…

Notice a pattern - the digits seem grouped together; using spaces to highlight the groups of digits,

1/998 001 = 000 001 002 003…009 010 011 012 013…

The 998 in this context is akin to -2, as with p-adics and 2's compliment, see below:

990 --> -10
995 --> -5
998 --> -2
999 --> -1
000 --> 0
012 --> 12
etc…

What's more, later on sequences can end up with many groups of trailing 0s that hold little meaning when looking at the sequence so I'll omit them. I'll also write 005 as just 5 and 998 as just -2.

This allows me to denote 1/99980001 as simply -2, 1. As the whole result is a rational number, more digits are required for the groups to result in longer sequences.

-2, 1 --> 1, 2, 3, 4, …
-2 --> 1, 2, 4, 8, 16, 32, …
-2, -1 --> 1, 1, 2, 3, 5, 8, … [name: Fibonacci sequence]
-2, -2 --> 1, 1, 3, 5, 11, 21, 43, … [name: sum of all previous terms, add an extra 1 every other term]

I think these sequences tend to be associated with the Fibonacci sequence, lines across the pascal triangle (and so N chooses K) and the golden ratio.

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Some general rules:

N represents the Nth value in the sequence and "|" means "where".

(starts > -1) --> not useful
(-1, < 1) --> not useful
(ends with 0) --> not useful

(P repeated K times) | (P < 0 && K > 1) -->
(sum * P + (1 every K otherwise 0)) or
(last * P + (sawtooth K long))

Specifically:

(P, P) | (P < 0) -->
(sum_all_previous_terms * (-P) + (1 on alternating terms)) or
(last_term * (-P) - 1 + (1 on alternating terms) * 2)

(P | P < 0) --> (-P)^N

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The following don't fit patterns specified in the above rules:

"tends" shows what the difference of adjacent terms tends towards.

-3, -2 --> 1, 2, 6, 16, 44, 120, 328, 896, 2448, … [tends: 2.73…]
-3, -1 --> 1, 2, 5, 12, 29, 70, 169, 408, … [tends: 2.414…]
-3, 1 --> 1, 3, 8, 21, 55, 144, … [tends: 2.6…]
-3, 2 --> 1, 3, 7, 15, 31, 63, … [name: 2^n - 1, tends: 2]
-3, 3 --> (same as -3, -3)
-2, -3 --> 1, 1, 4, 7, 19, 40, 97, 217, 508, 1159, 2683, 6160, 14209, 32689, … [tends: 2.3]
-2, -1 --> 1, 1, 2, 3, 5, 8, 13, … [name: Fibonacci sequence, tends: golden ratio]
-2, 1 --> 1, 2, 3, … [name: n, tends: 0]
-2, 2 --> 1, 2, 1, -1, -5, -9, -8, 0, 16, 32, 31, -1 -75, … [notes: absolute differences are 1, 1, 1, 4, 4, 1, 8, 16, 16, 1, 32, 74]
-2, 3 --> 1, 2, 0, -5, -12, -10, 13, 56, 72, -23, …
-1, 1 --> 1, 0, -1, -2, -1, 0 … [name: sawtooth wave]
-1, 2 --> 0, -2, -4, -1, 5, 6, -4, -18, …
-1, 3 --> 0, -3, -5, 1, 16, …

-2, 1, -2 --> see below
-2, 1, -4 --> see below

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My notes follow for two very strange sequences found, I cannot find them in the OEIS.

+pN and *N are guesses that it could be to do with adding the previous N terms or multiplying the previous term by N.

(-2, 1, -4) -->

1
2 — *1
2 — *1
4 — *2 or +p2
12 — *3
24 — *2
40 — +p3
80 — *2 or +p4
176
352 — *2
6721

Successive terms Coldatz lengths:

0, 1, 1, 2
9, 10
8, 9
18, 19
44

10 + 8 = 18
10 + 9 = 19
9 + 9 = 18

My notation of the "3.5th power of 2" here does not mean 2^3.5 but rather (2^3 + 2^4) / 2. In a similar way I wonder if things occurring only once every Nth term is to do with some influence having a sort of magnitude of 1/N.

(-2, 1, -2) -->

1 — 0th power of 2
2 2 2 — 1st power of 2
4 — 2nd power of 2
8 — 3rd power of 2
12 — "3.5th" power of 2
16 — 4th power of 2
24 — "4.5th" power of 2
40 — don't you mean 48? +p2
64 — 6th power of 2
96 — "6.5th" power of 2
144 — 96 * 1.5
224 — factors are just 2s and a 7
1984 — no, 1024?
 
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Country Boy

Math Team
Jan 2015
3,261
899
Alabama
\(\displaystyle 96= 32*3= 2^6*3\). In what sense is that the "'3.5th' power of 2"?

\(\displaystyle 2^{3.5}= 2^3*2^{0.5}= 8\sqrt{2}\) which is approximately 11.31.
 

Country Boy

Math Team
Jan 2015
3,261
899
Alabama
\(\displaystyle 96= 32*3= 2^6*3\). In what sense is that the "'3.5th' power of 2"?

\(\displaystyle 2^{3.5}= 2^3*2^{0.5}= 8\sqrt{2}\) which is approximately 11.31.
 
May 2016
1,310
551
USA
What do patterns in a particular representation of a number tell us about that number or some class of numbers, or am I missing the point?
 
Feb 2016
1,849
657
.
What do patterns in a particular representation of a number tell us about that number or some class of numbers, or am I missing the point?
You didn't go deep enough ;)
 

v8archie

Math Team
Dec 2013
7,703
2,675
Colombia
Before getting too excited about $\frac1{9801}$, I suggest looking at $\frac1{81}$
Note that $$0.01234567890123456789\ldots = \frac{13717421}{1111111111}$$