What is the pattern in the sequence 1 2 4 3 6 8 7 14 16?

Addison Parker 2022-09-11 Answered
What is the pattern in the sequence 1 2 4 3 6 8 7 14 16?
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Answers (1)

Karla Bautista
Answered 2022-09-12 Author has 16 answers
Not a very mathematically significant sequence, but can you express it algebraically with a single formula?
Consider ω = - 1 2 + i 3 2
This has the property that ω 3 = 1
Then we can write:
a 0 = 1
a i + 1 = ( ω i - ω ) ( ω i - ω 2 ) ( 1 - ω ) ( 1 - ω 2 ) 2 a i + ( ω i - ω 2 ) ( ω i - 1 ) ( ω - ω 2 ) ( ω - 1 ) ( a i + 2 ) + ( ω i - 1 ) ( ω i - ω ) ( ω 2 - 1 ) ( ω 2 - ω ) ( a i - 1 )
This can be simplified, but it helps to have it in this formulation so you can understand how it works.
When i=0 modulo 3, then:
( ω i - ω ) ( ω i - ω 2 ) ( 1 - ω ) ( 1 - ω 2 ) = ( 1 - ω ) ( 1 - ω 2 ) ( 1 - ω ) ( 1 - ω 2 ) = 1
( ω i - ω 2 ) ( ω i - 1 ) ( ω - ω 2 ) ( ω - 1 ) = ( 1 - ω 2 ) ( 1 - 1 ) ( 1 - ω 2 ) ( ω - 1 ) = 0
( ω i - 1 ) ( ω i - ω ) ( ω 2 - 1 ) ( ω 2 - ω ) = ( 1 - 1 ) ( 1 - ω ) ( ω 2 - 1 ) ( ω 2 - ω ) = 0
When i=1 modulo 3, then these coefficient expressions evaluate as 0, 1 and 0.
When i=2 modulo 3, then these coefficient expressions evaluate as 0, 0 and 1.
So we use these to pick out each of the three rules cyclically.

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