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 2017-05-15, 10:13 #1 MattcAnderson     "Matthew Anderson" Dec 2010 Oregon, USA 2×3×7×17 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, 12:57   #2
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

838410 Posts

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.

 2017-05-15, 15:08 #3 CRGreathouse     Aug 2006 32×5×7×19 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, 22:13 #4 MattcAnderson     "Matthew Anderson" Dec 2010 Oregon, USA 10110010102 Posts 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 Attached Thumbnails
 2017-05-15, 22:29 #5 MattcAnderson     "Matthew Anderson" Dec 2010 Oregon, USA 2·3·7·17 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
 2017-05-15, 22:32 #6 CRGreathouse     Aug 2006 10111011000012 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) 3×1,951 Posts
 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
 2017-05-20, 14:15 #9 MattcAnderson     "Matthew Anderson" Dec 2010 Oregon, USA 13128 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 32·5·7·19 Posts 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.
2017-05-21, 01:52   #11
paulunderwood

Sep 2002
Database er0rr

3,617 Posts

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"

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