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 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.
 August 9th, 2018, 05:53 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 $96= 32*3= 2^6*3$. In what sense is that the "'3.5th' power of 2"? $2^{3.5}= 2^3*2^{0.5}= 8\sqrt{2}$ which is approximately 11.31.
 August 9th, 2018, 05:54 AM #3 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 $96= 32*3= 2^6*3$. In what sense is that the "'3.5th' power of 2"? $2^{3.5}= 2^3*2^{0.5}= 8\sqrt{2}$ which is approximately 11.31.
 August 9th, 2018, 07:21 AM #4 Senior Member   Joined: May 2016 From: USA Posts: 1,310 Thanks: 551 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?
August 9th, 2018, 02:11 PM   #5
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 Originally Posted by JeffM1 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

August 9th, 2018, 02:44 PM   #6
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 Originally Posted by Joppy You didn't go deep enough
That's me. I'm so shallow as to be nothing but surface.

 August 9th, 2018, 03:06 PM #7 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,663 Thanks: 2643 Math Focus: Mainly analysis and algebra Before getting too excited about $\frac1{9801}$, I suggest looking at $\frac1{81}$ Note that $$0.01234567890123456789\ldots = \frac{13717421}{1111111111}$$

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