find the missing number

MattcAnderson · 2017-05-15, 10:13

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

science_man_88 · 2017-05-15, 12:57

Quote:

Originally Posted by MattcAnderson

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.

CRGreathouse · 2017-05-15, 15:08

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.

MattcAnderson · 2017-05-15, 22:13

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

MattcAnderson · 2017-05-15, 22:29

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

CRGreathouse · 2017-05-15, 22:32

Yes. The bases of the exponential expression are the roots of the characteristic equation of the recurrence, x^2 - 6x - 8.

henryzz · 2017-05-16, 06:56

https://en.wikipedia.org/wiki/Lucas_sequence

Harrywill · 2017-05-17, 05:15

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

Tell Me

MattcAnderson · 2017-05-20, 14:15

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

CRGreathouse · 2017-05-20, 22:13

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.

paulunderwood · 2017-05-21, 01:52

Quote:

Originally Posted by CRGreathouse

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"

2017-05-15, 10:13	#1
MattcAnderson "Matthew Anderson" Dec 2010 Oregon, USA 2⁵×5² Posts	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

2017-05-15, 22:13	#4
MattcAnderson "Matthew Anderson" Dec 2010 Oregon, USA 2⁵·5² Posts	Hi MersenneForum, Thanks to all the readers and participants. Also, thanks to the Maple command 'rsolve' the expression F(r)=2F(r-1)+4F(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 Attached Thumbnails

2017-05-15, 22:29	#5
MattcAnderson "Matthew Anderson" Dec 2010 Oregon, USA 2⁵×5² Posts	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 Attached Thumbnails

2017-05-17, 05:15	#8
Harrywill "Harry Willam" May 2017 USA 2²·5 Posts	What Number Should Replace the Question Mark? Option :- A) 9 B) 8 C) 6 D) 0 Tell Me Attached Thumbnails

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
Didn't even find a useless factor of an uninteresting number! Take that!	storflyt32	storflyt32	298	2018-09-27 02:36
Please help me find a composite number (test2)	allasc	Math	0	2010-12-27 13:37
What way would you find numbers with a set number of factors?	nibble4bits	Puzzles	18	2006-01-07 10:40
how do you find number of digits of a 2^n number?	Unregistered	Math	11	2004-11-30 22:53
Can you find the smallest number?	Fusion_power	Puzzles	8	2003-11-18 19:36

2017-05-15, 15:08	#3
CRGreathouse Aug 2006 13533₈ Posts	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.

2017-05-15, 22:32	#6
CRGreathouse Aug 2006 13533₈ Posts	Yes. The bases of the exponential expression are the roots of the characteristic equation of the recurrence, x^2 - 6x - 8.

2017-05-16, 06:56	#7
henryzz Just call me Henry "David" Sep 2007 Cambridge (GMT/BST) 2³×3×5×7² Posts	https://en.wikipedia.org/wiki/Lucas_sequence

2017-05-20, 14:15	#9
MattcAnderson "Matthew Anderson" Dec 2010 Oregon, USA 1440₈ Posts	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

2017-05-20, 22:13	#10
CRGreathouse Aug 2006 3·1,993 Posts	The characteristic equation of a recurrence of the form a(n) = Aa(n-1) + Ba(n-1) + Ca(n-2) + Da(n-3) is x^4 - Ax^3 - Bx^2 - Cx - D and you extend this to lower or higher degree in the obvious way.