My Math Forum  

Go Back   My Math Forum > Math Forums > Math

Math General Math Forum - For general math related discussion and news


Reply
 
LinkBack Thread Tools Display Modes
July 15th, 2018, 11:05 AM   #1
Newbie
 
Joined: Jul 2018
From: UK

Posts: 7
Thanks: 0

1/9801 and even deeper examples

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.

---

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

---

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

---

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?

Last edited by skipjack; July 15th, 2018 at 12:37 PM.
alan2here is offline  
 
August 9th, 2018, 05:53 AM   #2
Math Team
 
Joined: Jan 2015
From: Alabama

Posts: 3,261
Thanks: 894

. In what sense is that the "'3.5th' power of 2"?

which is approximately 11.31.
Country Boy is offline  
August 9th, 2018, 05:54 AM   #3
Math Team
 
Joined: Jan 2015
From: Alabama

Posts: 3,261
Thanks: 894

. In what sense is that the "'3.5th' power of 2"?

which is approximately 11.31.
Country Boy is offline  
August 9th, 2018, 07:21 AM   #4
Senior Member
 
Joined: May 2016
From: USA

Posts: 1,148
Thanks: 479

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?
JeffM1 is offline  
August 9th, 2018, 02:11 PM   #5
Senior Member
 
Joined: Feb 2016
From: Australia

Posts: 1,715
Thanks: 597

Math Focus: Yet to find out.
Quote:
Originally Posted by JeffM1 View Post
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
Joppy is offline  
August 9th, 2018, 02:44 PM   #6
Senior Member
 
Joined: May 2016
From: USA

Posts: 1,148
Thanks: 479

Quote:
Originally Posted by Joppy View Post
You didn't go deep enough
That's me. I'm so shallow as to be nothing but surface.
JeffM1 is offline  
August 9th, 2018, 03:06 PM   #7
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,445
Thanks: 2499

Math Focus: Mainly analysis and algebra
Before getting too excited about $\frac1{9801}$, I suggest looking at $\frac1{81}$
Note that $$0.01234567890123456789\ldots = \frac{13717421}{1111111111}$$
v8archie is offline  
Reply

  My Math Forum > Math Forums > Math

Tags
deeper, examples, generators, reciprocals, sequences



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
1/998001 goes way deeper than expected skipjack Math 1 July 16th, 2018 04:41 AM
Deeper Understanding of Limit Proofs Antoniomathgini Calculus 5 December 6th, 2017 08:41 PM
Examples... shaharhada Algebra 0 August 16th, 2012 05:55 AM
Examples in Lp and Hp spaces mattia90 Real Analysis 0 April 12th, 2012 07:03 AM
What are some examples of functions STV Real Analysis 1 July 3rd, 2008 02:29 PM





Copyright © 2018 My Math Forum. All rights reserved.