Sequence Numbers: Changing the Length of the Strings that the Decimal Expansion is Written with.

10/12/2015

Sequence Numbers: Changing the Length of the Strings that the Decimal Expansion is Written with.

Counting Sequences:

The Sequence Number 81 will produce single digit strings from zero to seven:

1/81 = 0.0123456790123…

The Sequence Number 9801 will produce two digit strings from zero to 97:

1/9801 = 0.000102030405060708091011112…

To continue lengthening the strings keep adding nines in front of the eight, and zeros in front of the one.  The length of the strings will be the same as the number of eight and nines, or the number of zeros and ones.  You should only have 1 eight and 1 one.  The number of nines should always be the same as the number of zeros.

Example: In order to produce strings that are 100 digits long you will need 99 nines, one eight, 99 zeros and 1 one.

Multiplication Sequences or Multiplication Tables:

Sequence Numbers for multiplication sequences must be one of the Sequence Numbers for counting sequences (81, 9,801, 998,001, etc.) divided by the number you want your multiplication sequence to multiply by (provided the numerator is divisible by the denominator).

All of the numbers used for the numerators are divisible by 3, 9, and 27 so any of the counting Sequence Numbers can be used to show multiples of three.  Each string will be written in ½ of the number of digits in the numerator.

Listed below are the counting Sequence Number, and their factors.

Let’s look as some of the other factors first:

Counting Sequence Numbers	Prime Factors
9,801	34 * 112
998,001	36 * 372
99,980,001	34 * 112 * 1012
9,999,800,001	34 * 412 * 2712
999,998,000,001	36 * 72 * 112 * 132 * 372
99,999,980,000,001	34 * 2392 * 46492
9,999,999,800,000,001	3^4 * 11^2 * 732 * 1012 * 1372
999,999,998,000,000,001	38 * 372 * 333,6672
99,999,999,980,000,000,001	32 * 112 * 412 * 2712 * 90912
9,999,999,999,800,000,000,001	34 * 216492 * 513,2392
999,999,999,998,000,000,000, 001	3^6 * 72 * 112 * 132 * 372 * 1012 * 99012
99,999,999,999,980,000,000, 000,001	34 * 532 * 792 * 265,371,6532
9,999,999,999,999,800,000, 000,000,001	34 * 112 * 2392 * 46492 * 909,0912
999,999,999,999,998,000,000, 000,000,001	36 * 312 * 372 * 412 * 2712 * 2,906,1612
99,999,999,999,999,980,000, 000,000,000,001	34 * 112 * 172 * 732 * 1012 * 1372 * 5,882,3532
9,999,999,999,999,999,800,000, 000,000,000,001	34 * 2,071,7232 * 5,363,222,3572
999,999,999,999,999,998,000, 000,000,000,000,001	38 * 72 * 112 * 132 * 192 * 372 * 525792 * 333,6672
99,999,999,999,999,999,980, 000,000,000,000,000,001	34 * 1,111,111,111,111,111,1112
9,999,999,999,999,999, 999,800,000,000,000,000, 000,001	34 * 112 * 412 * 1012 * 2712 * 35412 * 90912 * 279612
999,999,999,999,999,999, 998,000,000,000,000,000, 000,001	36 * 372 * 432 * 2392 * 19332 * 46492 * 10,838,6892
99,999,999,999,999,999, 999,980,000,000,000,000, 000,000, 001	34 * 114 * 232 * 40932 * 87792 * 216492 * 513,2392
9,999,999,999,999,999, 999,999,800,000,000,000, 000,000,000, 001	34 * 11,111,111,111,111,111,111, 1112
999,999,999,999,999,999, 999,998,000,000,000,000, 000,000, 000,001	36 * 72 * 112 * 132 * 372 * 732 * 1012 * 1372 * 99012 * 99,990,0012
99,999,999,999,999,999, 999,999,980,000,000,000, 000,000, 000,000,001	34 * 412 * 2712 * 214012 * 256012 * 182,521,213,0012
9,999,999,999,999,999, 999, 999,999,800,000,000, 000,000,000,000,000,001	34 * 112 * 532 * 792 * 8592 * 2653716532 * 10583130492
999,999,999,999,999,999, 999,999,998,000,000,000, 000,000,000,000,000,001	310 * 372 * 7572 * 3336672 * 4403346547776312
99,999,999,999,999, 999,999,999,999,980, 000,000,000,000,000, 000,000,000,001	34 * 112 * 292 * 1012 * 2392 * 2812 * 46492 * 9090912 * 1214994492
9,999,999,999,999,999, 999,999,999,999,800,000, 000,000,000,000,000,000, 000,001	34 * 31912 * 167632 * 430372 * 620032 * 778438393972
999,999,999,999,999, 999,999,999,999,998, 000,000,000,000,000, 000,000,000,000,001	36 * 72 * 112 * 132 * 312 * 372 * 412 * 2112 * 2412 * 2712 * 21612 * 90912 * 29061612

Notice that 9,801 cannot be used to produce a Sequence Number showing the multiples of seven because it is not divisible by seven.

999,998,000,001 can be used to produce a Sequence Number showing multiples of seven because 999,998,000,001 is divisible by seven.

999,998,000,001/7 = 142,856,857,143 so 142,856,857,143 is one possible sequence number that will produce a decimal expansion showing multiples of 7 (in this case the terms will be written in six digit strings).

999,999,999,998,000,000,000,001/13 will produce the smallest Sequence Number that produces a list of the multiples of 13, because 999,999,999,998,000,000,000,001 is the smallest of the counting Sequence Number that is divisible by 13.  By the way, 999,999,999,998,000,000,000,001/13 is 76923,076,922,923,076,923,077, which is a Sequence Number that shows multiples of 13 written in 12 digit strings.

All of the Sequence Numbers used to produce counting sequences end in the digit one, so none of them are divisible by 2 or 5.  If you want to find a Sequence Number to use with a number that is a multiple of 2 or 5 you will have to add zeros to the end of that Sequence Number.  The number of zeros should be half the length of the Sequence Number.

So 998,001,000 can be used to show multiples of 2, 4, 5, 8, 10, 20, 25, 40 and 125, along with any combinations of the multiples of 2, 3, 5, or 37 that divide 998,001,000 evenly.

It you have to add zeros, please remember that the string length will be half the length of the counting sequence number that you start with, then add that number of zeros to the counting sequence number.

Power Sequences:

For Power Sequence the length of the strings in the decimal expansion will be equal to the length of the Sequence Number.

Example: The Sequence Number 999,995 will show powers of five listed in six digit strings (beginning with 5⁰).

1/999995 = 0.000000000005000025000125000615 …

Combinatoric (or Combinatorial) Functions:

C (n, m) or the number of ways to choose m items from a group of items.

The Sequences Numbers representing C (n, m) will consist of some number of nines equal in length to the length of the strings you desire,  and raised to the m+1 power. 

Example: The Sequence Number equal to 999⁶ will produce the terms for C (n, 5) writing in three digit strings.

999⁶ = 994,014,980,014,994,001.

1/994014980014994001 =

0.
000  000  000  000  000  001  006  021  056  126  …

Fibonacci Like Sequences:

If you will recall:

The Sequence Number for the Fibonacci Sequence and related sequences are composed of two parts.  To produce n digit sequences each part has to be written with n digits.

Example: The Sequence Number 999,998,999,999 produces the Fibonacci Sequence written in six digit strings.

The Sequence Number for the Tribonacci Sequence and related sequences are composed of three parts.  To produce n digit strings each part has to be n digit parts.

Example: The Sequence Number 999,999,998,999,999.998,999,999,999 produces the Tribonacci Sequence written in nine digit strings.

Sequence Numbers for Tetranacci like sequences must have four parts, and the length of each part must be equal to each other.  The number of digits in each part will be the number of digits in each string in the decimal expansion.

This pattern continues for other Fibonacci like sequences (Pentanacci, Hexanacci, etc.)

David