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Old 2017-05-15, 10:13   #1
MattcAnderson
 
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"Matthew Anderson"
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Default find the missing number

Hi MersenneForum,

I was doodling with a spreadsheet program. Can anyone find the missing number in this sequence?

1,4,32,224,1600,?,81152,578048, ...

Also, what is the general form that will produce more numbers in the sequence?

Best of luck.
Regards,
Matt

There are two arbitrary initial values
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Old 2017-05-15, 12:57   #2
science_man_88
 
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Quote:
Originally Posted by MattcAnderson View Post
Hi MersenneForum,

I was doodling with a spreadsheet program. Can anyone find the missing number in this sequence?

1,4,32,224,1600,?,81152,578048, ...

Also, what is the general form that will produce more numbers in the sequence?

Best of luck.
Regards,
Matt

There are two arbitrary initial values
all I've got is all the numbers other than one shown are multiples of 4 ( in fact all the one shown greater than 4 are also multiples of 32), the sequence of differences repeatedly applied gives that if they are expressible as a polynomial it's first term must be of degree 4 or greater ( likely greater) I haven't tried CRG's fitExp and other codes on them. in theory other than they are said to follow a pattern it could be any number.
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Old 2017-05-15, 15:08   #3
CRGreathouse
 
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The missing number is 11392 and the formula is a(n) = 6a(n-1) + 8a(n-2). That also explains why the number of factors of 2 tends to increase.
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Old 2017-05-15, 22:13   #4
MattcAnderson
 
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Hi MersenneForum,

Thanks to all the readers and participants.

Also, thanks to the Maple command 'rsolve' the expression

F(r)=2*F(r-1)+4*F(r-2)

can be expressed as shown in the second attachment.

Let me know if a .png file is no good for you.

Regards,
Matt
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Old 2017-05-15, 22:29   #5
MattcAnderson
 
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Hi Mersenneforum,

Some of you may be familiar with the radical expression that goes with the Fibonacci sequence.

https://en.wikipedia.org/wiki/Fibona...e_golden_ratio

This should be similar.

Regards,
Matt
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Old 2017-05-15, 22:32   #6
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Yes. The bases of the exponential expression are the roots of the characteristic equation of the recurrence, x^2 - 6x - 8.
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Old 2017-05-16, 06:56   #7
henryzz
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https://en.wikipedia.org/wiki/Lucas_sequence
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Old 2017-05-17, 05:15   #8
Harrywill
 
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Default What Number Should Replace the Question Mark?

Option :-
A) 9
B) 8
C) 6
D) 0

Tell Me
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Old 2017-05-20, 14:15   #9
MattcAnderson
 
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Hi Mersenneforum,

Many thanks to the responses so far. As a learning exercise it would be nice to know the steps between a 'recurrence relation of the first kind' and its 'characteristic equation'.

From the Wikipedia article on Lucas Sequence, we see this -

Explicit expressions,

The characteristic equation of the recurrence relation for Lucas sequences

x^2 - Px + Q = 0

Must go.

Matt
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Old 2017-05-20, 22:13   #10
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The characteristic equation of a recurrence of the form

a(n) = A*a(n-1) + B*a(n-1) + C*a(n-2) + D*a(n-3)

is

x^4 - Ax^3 - Bx^2 - Cx - D

and you extend this to lower or higher degree in the obvious way.
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Old 2017-05-21, 01:52   #11
paulunderwood
 
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Quote:
Originally Posted by CRGreathouse View Post
The characteristic equation of a recurrence of the form

a(n) = A*a(n-1) + B*a(n-1) + C*a(n-2) + D*a(n-3)

is

x^4 - Ax^3 - Bx^2 - Cx - D

and you extend this to lower or higher degree in the obvious way.
I am just nit-picking: "x^4 - Ax^3 - Bx^2 - Cx - D" is not an equation. I think you meant "x^4 - Ax^3 - Bx^2 - Cx - D = 0"
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