## 5. Remarkable patterns

### 123...

```1/81 = 0.012345679 ... (from 0 to 7 (one letter), last is 9. length=9)
1/891 = 0.001122334455667789 ... (from 00 to 77 (two letters), last is 89. length=18)
1/8991 = 0.000111222333444555666777889 ... (from 000 to 777 (three letters), last is 889. length=27)
1/89991 = 0.000011112222333344445555666677778889 ... (from 0000 to 7777 (four letters), last is 8889. length=36)
...

1/81 = 0.012345679 ... (length=9)
1/8181 = 0.000122234445666788901112333455567779 ... (3 letters. length=36)
1/818181 = 0.000001222223444445666667888890111112333334555556777779 ... (5 letters. length=54)
1/81818181 = 0.000000012222222344444445666666678888889011111112333333345555555677777779 ... (7 letters. length=72)
...

1/9801 = 0.000102030405060708091011121314151617181920212223242526272 ... 9799 ... (from 00 to 99 (2 digits) except 98, length=198)
1/998001 = 0.000001002003004005006007008009010011012013014015016017018 ... 997999 ... (from 000 to 999 (3 digits) except 998, length=2997)
1/99980001 = 0.000000010002000300040005000600070008000900100011001200130 ... 99979999 ... (from 0000 to 9999 (4 digits) except 9998, length=39996)
...
```

### Power of n

```1/49 = 0.02040816326530612244897959183673469387755 ...
```

Power of 2 appears.

``````
0.02040816326530612244897959183673469387755 ...
2 | | | | |
| | | | |
8 | | |
16 | |
32 |
65
``````

At a first glance, the pattern corrupts at 65, but

``````
64
128
| 256
| | 512
| |  1024
| |  | 2048
....         | |  | |
---------------------------
0.020408163265306122448979
``````

so the pattern continues eternaly.

[general formula]

For power of "a" in k-digits,

a/(10^k-a)

``````1/49 = 0.020408163265306122448979591836 ...
1/499 = 0.002004008016032064128256513026 ...
1/4999 = 0.000200040008001600320064012802 ...
1/49999 = 0.000020000400008000160003200064 ...
1/499999 = 0.000002000004000008000016000032 ...
1/4999999 = 0.000000200000040000008000001600 ...
1/49999999 = 0.000000020000000400000008000000 ...
1/499999999 = 0.000000002000000004000000008000 ...
1/4999999999 = 0.000000000200000000040000000008 ...

3/97 = 0.030927835051546391752577319587 ...
3/997 = 0.003009027081243731193580742226 ...
3/9997 = 0.000300090027008102430729218765 ...
3/99997 = 0.000030000900027000810024300729 ...
3/999997 = 0.000003000009000027000081000243 ...
3/9999997 = 0.000000300000090000027000008100 ...
3/99999997 = 0.000000030000000900000027000000 ...
3/999999997 = 0.000000003000000009000000027000 ...
3/9999999997 = 0.000000000300000000090000000027 ...

1/24 = 0.041666666666666666666666666666 ...
1/249 = 0.004016064257028112449799196787 ...
1/2499 = 0.000400160064025610244097639055 ...
1/24999 = 0.000040001600064002560102404096 ...
1/249999 = 0.000004000016000064000256001024 ...
1/2499999 = 0.000000400000160000064000025600 ...
1/24999999 = 0.000000040000001600000064000002 ...
1/249999999 = 0.000000004000000016000000064000 ...
1/2499999999 = 0.000000000400000000160000000064 ...

1/19 = 0.052631578947368421 ... (18 digits)
1/199 = 0.005025125628140703517587939698 ...
1/1999 = 0.000500250125062531265632816408 ...
1/19999 = 0.000050002500125006250312515625 ...
1/199999 = 0.000005000025000125000625003125 ...
1/1999999 = 0.000000500000250000125000062500 ...
1/19999999 = 0.000000050000002500000125000006 ...
1/199999999 = 0.000000005000000025000000125000 ...
1/1999999999 = 0.000000000500000000250000000125 ...

3/47 = 0.063829787234042553191489361702 ...
3/497 = 0.006036217303822937625754527162 ...
3/4997 = 0.000600360216129677806684010406 ...
3/49997 = 0.000060003600216012960777646658 ...
3/499997 = 0.000006000036000216001296007776 ...
3/4999997 = 0.000000600000360000216000129600 ...
3/49999997 = 0.000000060000003600000216000012 ...
3/499999997 = 0.000000006000000036000000216000 ...
3/4999999997 = 0.000000000600000000360000000216 ...

7/93 = 0.075268817204301075268817204301 ...
7/993 = 0.007049345417925478348439073514 ...
7/9993 = 0.000700490343240268187731411988 ...
7/99993 = 0.000070004900343024011680817657 ...
7/999993 = 0.000007000049000343002401016807 ...
7/9999993 = 0.000000700000490000343000240100 ...
7/99999993 = 0.000000070000004900000343000024 ...
7/999999993 = 0.000000007000000049000000343000 ...
7/9999999993 = 0.000000000700000000490000000343 ...

2/23 = 0.0869565217391304347826 ... (22 digits)
1/124 = 0.008064516129032258064516129032 ...
1/1249 = 0.000800640512409927942353883106 ...
1/12499 = 0.000080006400512040963277062164 ...
1/124999 = 0.000008000064000512004096032768 ...
1/1249999 = 0.000000800000640000512000409600 ...
1/12499999 = 0.000000080000006400000512000040 ...
1/124999999 = 0.000000008000000064000000512000 ...
1/1249999999 = 0.000000000800000000640000000512 ...

9/91 = 0.098901 ... (6 digits)
9/991 = 0.009081735620585267406659939455 ...
9/9991 = 0.000900810729656691021919727754 ...
9/99991 = 0.000090008100729065615905431488 ...
9/999991 = 0.000009000081000729006561059049 ...
9/9999991 = 0.000000900000810000729000656100 ...
9/99999991 = 0.000000090000008100000729000065 ...
9/999999991 = 0.000000009000000081000000729000 ...
9/9999999991 = 0.000000000900000000810000000729 ...
``````

In case of 11,

``` 11/89 = 0.123595505617977528 ... ```

This is,

``````
0.123595505617977528
1   | | |
2  | | |
3 | | |
5  | |
8 | |
13  |
21 |
34
55
89
...
``````

so the expansion becomes Fibonacci number.

### Multiple

Using gereration function.
Let s be a formal power series which coefficient is the multiple of k,

``````s = kx + 2kx^2 + 3kx^3 + 4kx^4 + ...
= k(x + 2x^2 + 3x^3 + 4x^4 ...)
``````

So, start from s = x + 2x^2 + 3x^3 + 4x^4 ...

``````       sx = x^2 + 2x^3 + 3x^4 + 4x^5 ...
∴ (1-x)s = x + x^2 + x^3 + ...
``````

The right side is a geometric series which first term is x and ratio is x, so,

``````      = x/(1-x)
∴ s = x/(1-x)^2
``````

Let x replace by 1/x then,

``` s = x/(x-1)^2 ```

General formula is,

``` 2 digits : k*100/99^2 3 digits : k*1000/999^2 4 digits : k*10000/9999^2 ... ```

We can transform this for unit fraction, if we don't adhere to the start of first digit,

``````multiple of 2
2/99^2 ⇒ 1/(5*99^2)=1/49005=0.000020406081012141618202224262 ...
2/999^2 ⇒ 1/(5*999^2)=1/4990005=0.000000200400600801001201401601 ...
2/9999^2 ⇒ 1/(5*9999^2)=1/499900005=0.000000002000400060008001000120 ...
...

multiple of 4
4/99^2 ⇒ 1/(25*99^2)=1/245025=0.000004081216202428323640444852 ...
4/999^2 ⇒ 1/(25*999^2)=1/24950025=0.000000040080120160200240280320 ...
4/9999^2 ⇒ 1/(25*9999^2)=1/2499500025=0.000000000400080012001600200024 ...
...

multiple of 8
8/99^2 ⇒ 1/(125*99^2)=1/1225125=0.000000816243240485664728088970 ...
8/999^2 ⇒ 1/(125*999^2)=1/124750125=0.000000008016024032040048056064 ...
8/9999^2 ⇒ 1/(125*9999^2)=1/12497500125=0.000000000080016002400320040004 ...
...

multiple of 5
5/99^2 ⇒ 1/(2*99^2)=1/19602=0.000051015202530354045505560657 ...
5/999^2 ⇒ 1/(2*999^2)=1/1996002=0.000000501001502002503003504004 ...
5/9999^2 ⇒ 1/(2*9999^2)=1/199960002=0.000000005001000150020002500300 ...
...

multiple of 3
3/99^2=1/33*99=1/3267=0.000306091215182124273033363942 ...
3/999^2=1/333*999=1/332667=0.000003006009012015018021024027 ...
3/9999^2=1/3333*9999=1/33326667=0.000000030006000900120015001800 ...
...

multiple of 9
9/99^2=1/11*99=1/1089=0.0009182736455463728191 ... (22 digits)
9/999^2=1/111*999=1/110889=0.000009018027036045054063072081 ...
9/9999^2=1/1111*9999=1/11108889=0.000000090018002700360045005400 ...
...

multiple of 6
6/99^2=2/33*99=10/5*33*99 ⇒ 1/16335=0.000061218243036424854606672788 ..
6/999^2=2/333*999=10/5*333*999 ⇒ 1/1663335=0.000000601201802403003604204805 ...
6/9999^2=2/3333*9999=10/5*3333*9999 ⇒ 1/166633335=0.000000006001200180024003000360 ...
...
``````

In case of multiple of 7,

``````7/999999^2=1/142857*999999=1/142856857143
7/999999999999^2=1/142857142857*999999999999=1/142857142856857142857143
7/999999999999999999^2=1/142857142857142857*999999999999999999=1/142857142857142856857142857142857143
...

1/142856857143 = 0.000000000007000014000021000028000035000042000049000056 ...
1/142857142856857142857143 = 0.00000000000000000000000700000000001400000000002100000000002800000000003500000000 ...
1/142857142857142856857142857142857143 = 0.00000000000000000000000000000000000700000000000000001400000000000000002100000000 ...
...
``````

Multiple can be not only 1 digit.

``````multiple of 11
11/99^2=1/9*99=1/891
111/999^2=1/9*999=1/8991
1111/9999^2=1/9*9999=1/89991
...
``````

[Appendix] Odd pattern, 0.00010305070911 ... is represented as

(1+x)/(1-x)^2 ⇒ x(x+1)/(x-1)^2

``````11/9^2=0.1358024691358024691358024691358024691358024691357 ...
101/99^2=0.01030507091113151719212325272931333537394143454749 ...
1001/999^2=0.001003005007009011013015017019021023025027029031032 ...
10001/9999^2=0.0001000300050007000900110013001500170019002100230024 ...
100001/99999^2=0.00001000030000500007000090001100013000150001700018999 ...
1000001/999999^2=0.000001000003000005000007000009000011000013000015000016 ...
10000001/9999999^2=0.000000100000030000005000000700000090000011000001300000 ...
``````

### 1/243

which is introduced in "Surely you are joking, Mr. Feynmann".

``````1/243 = 0.004115226337448559670781893 ... (length=27)
1/243243 = 0.000004111115222226333337444448555559666670777781888893 ... (length=54)
1/243243243 = 0.000000004111111115222222226333333337444444448555555559666666670777777781888888893 ... (length=81)
..
``````

In general, this pattern sometimes appears at the multiple of 9 by repeating sequence of inverse of prime. (243 = 9 * 027, 027 is the repeating sequence of 1/37).
So let's check the factors of repunits.

``````
digits | factors
-------------
3 : 3, 37
4 : 11, 101
5 : 41, 271
6 : 7, 13
7 : 239, 4649
8 : 73, 137
9 : 333667
10 : 9091
11 : 21649, 513239
12 : 9901
13 : 53, 79, 265371653
14 : 909091
15 : 31, 2906161
16 : 17, 5882353
17 : 2071723, 5363222357
18 : 19, 52579
19 : 1111111111111111111
20 : 3541, 27961
``````

For example,

``````101 : 99 : 1/(99*9)=1/891=
0.001122334455667789 ...(18 digits)

37 : 27 : 1/(27*9)=1/243=
0.004115226337448559670781893 ...(27 digits)

41 : 2439 : 1/(2439*9)=1/21951=
0.000045556011115666712222677782333378889344449 ...(45 digits)

7 : 142857 : 1/(142857*9)=1/1285713=
0.000000777778555556333334111111888889666667444445222223 ...(54 digits)

13 : 76923 : 1/(76923*9)=1/692307=
0.000001444445888890333334777779222223666668111112555557 ...(54 digits)

73 : 1369863 : 1/(1369863*9)=1/12328767=
0.000000081111111922222230333333414444445255555563666666747777778588888897 ...(72 digits)

271 : 369 : 1/(369*9)=1/3321=
0.000301114122252333634447455585666967780788919 ...(45 digits)

239 : 41841 : 1/(41841*9)=1/376569=
0.000002655555821111137666669322222487777804333335988889154444471 ...(63 digits)

137 : 729927 : 1/(729927*9)=1/6569343=
0.000000152222223744444459666666818888890411111126333333485555557077777793 ...(72 digits)

53 : 188679245283 : 1/(188679245283*9)=1/1698113207547=
0.000000000000588888888888947777777777783666666666667255555555555614444444444450333333333333922222222222281111111111117 ...(117 digits)

79 : 126582278481 : 1/(126582278481*9)=1/1139240506329=
0.000000000000877777777777865555555555564333333333334211111111111198888888888897666666666667544444444444532222222222231 ...(117 digits)

31 : 32258064516129 : 1/(32258064516129*9)=1/290322580645161=
0.000000000000003444444444444447888888888888892333333333333336777777777777781222222222222225666666666666670111111111111114555555555555559 ...(135 digits)

17 : 588235294117647 : 1/(588235294117647*9)=1/5294117647058823=
0.000000000000000188888888888888907777777777777779666666666666666855555555555555574444444444444446333333333333333522222222222222241111111111111113 ...(144 digits)

19 : 52631578947368421 : 1/(52631578947368421*9)=1/473684210526315789=
0.000000000000000002111111111111111113222222222222222224333333333333333335444444444444444446555555555555555557666666666666666668777777777777777779888888888888888891 ...(162 digits)

3541 : 28240609997175939 : 1/(28240609997175939*9)=1/254165489974583451=
0.000000000000000003934444444444444444483788888888888888889282333333333333333337267777777777777777817122222222222222222615666666666666666670601111111111111111150455555555555555555949 ...(180 digits)

27961 : 3576409999642359 : 1/(3576409999642359*9)=1/32187689996781231=
0.000000000000000031067777777777777778088455555555555555558662333333333333333364401111111111111111421788888888888888891995666666666666666697734444444444444444755122222222222222225329 ...(180 digits)

9091 : 1099989 : 1/(1099989*9)=1/9899901=
0.000000101011111121212222223232333333434344444454545555556565666666767677777787878888889899 ...(90 digits)

9901 : 100999899 : 1/(100999899*9)=1/908999091=
0.000000001100111111112211222222223322333333334433444444445544555555556655666666667766777777778877888888889989 ...(108 digits)

909091 : 109999989 : 1/(109999989*9)=1/989999901=
0.000000001010101111111121212122222222323232333333334343434444444454545455555555656565666666667676767777777787878788888888989899 ...(126 digits)
``````

### Fibonacci number, Lucas number

``````
Fibonacci number : 0, 1, 1, 2, 3, 5, ...
Lucas number     2, 1, 3, 4, 7, 11, ...
``````

Using generation function

``````
F = x + x^2 + 2x^3 + 3x^4 + 5x^5 + ...
x.F =     x^2 +  x^3 + 2x^4 + 3x^5 + ...
x^2.F =            x^3 +  x^4 + 2x^5 + ...

F - x.F - x^2.F = x
∴ F = x/(1-x-x^2)
``````

Let x replace by 1/x,

```  (1/x)/{1-1/x-1/x^2} =(1/x)/{(x^2-x-1)/x^2} =x/(x^2-x-1) 10/(10^2-10-1)=10/89=0.112359550561797752808988764044 ... 100/(100^2-100-1)=100/9899=0.010102030508132134559046368320 ... 1000/(1000^2-1000-1)=1000/998999=0.001001002003005008013021034055 ... ```

If we don't adhere the start decimal digit, then,

1/89, 1/9899, 1/998999, ...

The generation function of Lucas number is,

``````
L = 2 + x + 3x^2 + 4x^3 + 7x^4 + ...
x.L =    2x +  x^2 + 3x^3 + 4x^4 + ...
x^2.L =         2x^2 +  x^3 + 3x^4 + ...

(1-x-x^2)L=2-x
∴ L=(2-x)/(1-x-x^2)
``````

Let x replace by 1/x,

```  (2-1/x)/(1-1/x-1/x^2) ={(2x-1)/x}/{(x^2-x-1)/x^2} =x(2x-1)/(x^2-x-1) 10(20-1)/(10^2-10-1)=190/89=2.1348314606741573033707865168 ... 100(200-1)/(100^2-100-1)=19900/9899=2.0103040711182947772502272956 ... 1000(2000-1)/(1000^2-1000-1)=1999000/998999=2.0010030040070110180290470761 ... ```

Multiply 99...99 for transform into unit fraction,

``` 19 : 18 digits : 19/89*(10^18-1)=1/4684210526315789469 199 : 99 digits 1999 : 999 digits 1/4684210526315789469=0.000000000000000000213483146067... ```

### Power series

Case of power 2

`````` s = 1 + 4x + 9x^2 + 16x^3 + ...
xs =      x + 4x^2 +  9x^3 + ...

a=s-xs = 1 + 3x + 5x^2 + 7x^3 + ...
xa =      x + 3x^2 + 5x^3 + ...

a-xa = 1 + 2x + 2x^2 + 2x^2 + ...

(1-x)a-1 : first term 2x, ratio x
= 2x/(1-x)
a = {2x/(1-x) + 1}/(1-x)
= (1+x)/(1-x)^2

s=(1+x)/(1-x)^3
``````

Let x replace by 1/x,

```  (1+1/x)/(1-1/x)^3 ={(1+x)/x}/{(x-1)/x}^3 =x^2(x+1)/(x-1)^3 100^2*101/99^3= 1.040916253649648201224570982758 ... 1000^2*1001/999^3= 1.004009016025036049064081100121 ... 10000^2*10001/9999^3= 1.000400090016002500360049006400 ... 100000^2*100001/99999^3= 1.000040000900016000250003600049 ... 1000000^2*1000001/999999^3= 1.000004000009000016000025000036 ... ```

transform into unit fraction,

``````100 : 101/9999*99^3=1/96059601
1000 : 1001/999999*999^3=1/996005996001
10000 : 10001/99999999*9999^3=1/9996000599960001
``````

For higher power,

``````Power 3 : (x^2+4x+1)/(1-x)^4 ⇒ x^2.(x^2+4x+1)/(x-1)^4
1000^2*1004001/999^4= 1.008027064125216343512730001332730199747379100918 ...
10000^2*100040001/9999^4= 1.000800270064012502160343051207291000133117282197 ...
100000^2*10000400001/99999^4= 1.000080002700064001250021600343005120072901000012 ...
1000000^2*1000004000001/999999^4= 1.000008000027000064000125000216000343000512000728 ...

Power 4 : (x^3+11x^2+11x+1)/(1-x)^5 ⇒ x^2.(x^3+11x^2+11x+1)/(x-1)^5
1000^2*1011011001/999^5= 1.016081256626298405102571014661764599466690619626 ...
10000^2*1001100110001/9999^5= 1.001600810256062512962401409665620001464307388564 ...
100000^2*1000110001100001/99999^5= 1.000160008100256006250129602401040960656110000145 ...
1000000^2*1000011000011000001/999999^5= 1.00001600008100025600062500129600240100409600656 ...
10000000^2*1000001100000110000001/9999999^5= 1.000001600000810000256000062500012960002401000409 ...

Power 5 : (x^4+26x^3+66x^2+26x+1)/(1-x)^6 ⇒ x^2.(x^4+26x^3+66x^2+26x+1)/(x-1)^6
10000^2*10026006600260001/9999^6= 1.003202431024312577776810277390590016107588691346 ...
100000^2*100026000660002600001/99999^6= 1.00032002430102403125077761680732768590500000161 ...
1000000^2*1000026000066000026000001/999999^6= 1.000032000243001024003125007776016807032768059048 ...
10000000^2*10000026000006600000260000001/9999999^6= 1.000003200002430001024000312500077760016807003276 ...
100000000^2*100000026000000660000002600000001/99999999^6= 1.000000320000024300001024000031250000777600016806 ...
``````

### A smalltalk : Reciprocal patterns

``` 1/27=0.037037037 ... (repetition of 37) 1/37=0.027027027 ... (repeticion of 27) ```

previous index

E-mail : kc2h-msm@asahi-net.or.jp
Hisanori Mishima