Number Patterns

Trigonometric Expansion

Use the Maple command expand to generate the following formulae:


MATH

MATH

MATH

MATH

MATH

MATH
The coefficients can be arranged in a table

1                  
  1                
-1 0 2 0 0 0 0 0 0 0
0 -3 0 4 0 0 0 0 0 0
1 0 -8 0 8 0 0 0 0 0
0 5 0 -20 0 16 0 0 0 0
-1 0 18 0 -48 0 32 0 0 0
0 -7 0 56 0 -112 0 64 0 0
1 0 -32 0 160 0 -256 0 128 0
0 9 0 -120 0 432 0 -576 0 256

How is this table generated?

Expressing $\cos ^{n}\theta $ in Terms of $\cos k\theta $

Use the Maple command combine to generate the following formulae:


MATH

MATH

MATH

MATH

MATH

MATH

MATH

The coefficients can be arranged in a table

  1                
1 0 1 0 0 0 0 0 0 0
0 3 0 1 0 0 0 0 0 0
3 0 4 0 1 0 0 0 0 0
0 10 0 5 0 1 0 0 0 0
10 0 15 0 6 0 1 0 0 0
0 35 0 21 0 7 0 1 0 0
35 0 56 0 28 0 8 0 1 0
0 126 0 84 0 36 0 9 0 1

How is this table generated?

Successive Derivatives of $\frac{e^{x}}{1+x}$

Define MATH Then
MATH

MATH

MATH

MATH

MATH

MATH
The coefficients can be arranged in a table

1              
1 -1 0 0 0 0 0 0
1 -2 2 0 0 0 0 0
1 -3 6 -6 0 0 0 0
1 -4 12 -24 24 0 0 0
1 -5 20 -60 120 -120 0 0
1 -6 30 -120 360 -720 720 0
1 -7 42 -210 840 -2520 5040 -5040

How is this table generated?

Same Formula, Different Pattern


MATH

MATH

MATH

MATH

MATH

It is clear that in general we have
MATH
for some polynomial function $p_{n}(x)$ of degree $n.$ Indeed from

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we see that
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for $n\ge 0.\,$Thus we have the recurrence relations
MATH

MATH
for $n\ge 0.$

The following table is generated in view of these relations:

1            
0 1 0 0 0 0 0
1 0 1 0 0 0 0
-2 3 0 1 0 0 0
9 -8 6 0 1 0 0
-44 45 -20 10 0 1 0
265 -264 135 -40 15 0 1
-1854 1855 -924 315 -70 21 0


 

Factors of $x^{2n}+x^{n}+1$

Applying the Factor command, we have


MATH

MATH

MATH

MATH

Can you see the general pattern?