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The “Look and Say Sequence” is not a typical numerical sequence.
Examples of typical numerical sequences are: arithmetic progression, geometric progression, Fibonacci sequence, etc.
However, although they appear to be at first glance, none of three examples shown below are numerical sequences.
All three examples fall into a class of NON-MATHEMATICAL progressions known as a “LOOK and SAY SEQUENCE”.
Example 1: Look and Say Sequence
1
11
21
1211
111221
312211
?
Example 2: Look and Say Sequence
2
12
1112
3112
132112
1113122112
?
Example 3: Look and Say Sequence
99
29
1219
11121119
31123119
132112132119
?
The nth value in these progressions is not computed using a mathematical formula. Instead, every number in the sequence is determined by reading the n+1 value “out loud”.
The “Look and Say Sequence” can be demonstrated using the first example:
(Note: an algorithm can be developed mimic this process.)
1
11
21
1211
111221
312211
?
Read any of the numbers out loud. This will describe the previous entry in the sequence.
With the exception of the first entry, every value consists of “pairs of numbers”.
For example, row 5 consists of the following three pairs of numbers: “11”, “12”, and “21”.
The first number in each pair describes how many times the second number is repeated in the previous number in the sequence.
To find the fourth number in the sequence, read the pairs of numbers in fifth number in the sequence, one pair at a time.
11 should be read “one 1”
12 should be read “one 2”
21 should be read “two 1’s”
Fourth (the previous number) is: 1211
The entire sequence can be developed as follows
1
This is the beginning of the sequence. It is not necessary to read the 1st row aloud.
11
Read the pairs of numbers in the 2nd row out loud to describe the 1st (previous) row.
The first number in each pair tells you how many times the second number in the pair appears in the previous row.
Number pairs: “11”
Read as: “One 1”
Row one (the first number in the sequence) is: 1
21
Read the pairs of numbers in the 3rd row out loud to describe the 2nd (previous) row.
The first number in each pair tells you how many times the second number in the pair appears in the previous row.
Number pairs: “21”
Read as: “Two 1’s”
Row two (the second number in the sequence) is: 11
1211
Read the pairs of numbers in the 4th row out loud to describe the 3rd (previous) row.
The first number in each pair tells you how many times the second number in the pair appears in the previous row.
Number pairs: “12”, and “11”
Read as: “One 2, One 1”
Row three (the third number in the sequence) is: 21
111221
Read the pairs of numbers in the 5th row out loud to describe the 4th (previous) row.
The first number in each pair tells you how many times the second number in the pair appears in the previous row.
Number pairs: “11”, “12”, and “21”
Read as: “One 1, One 2, Two 1’s”
Row four (the fourth number in the sequence) is: 1211
312211
Read the pairs of numbers in the 6th row out loud to describe the 5th (previous) row.
The first number in each pair tells you how many times the second number in the pair appears in the previous row.
Number pairs: “31”, “22”, and “11”
Read as: “Three 1’s, One 2’s, One 1”
Row five (the fifth number in the sequence) is: 111221
7th row: SEQUENCE UNKNOWN
work backwards
The sequence in the 6th (previous) row is known
Sixth (previous) row: 312211
Therefore, the verbal description of the 7th row should be:
Read as: “One 3”, “One 1”, “Two 2’s”, “Two 1’s”
Therefore, the number sequence for the 7th row is: 13 11 22 21
The sequence, including the 7th value, is:
1
11
21
1211
111221
312211
13112221…