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Help deriving integer sequence formula

by ktoz
Tags: deriving, formula, integer, sequence
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ktoz
#1
Jan16-10, 03:51 PM
P: 145
Hi

I'm playing around with partitions and have come up with an integer sequence representing the maximum number of partitions of various "widths" that display the following properties:

- min values in partition are equal
- max values in partition are equal
- partitions contain equal number of members
- sum of members is equal

For example, given:

min = 1
max = 6
count = 4
sum = 14

There are only two partitions that satisfy the constraints

{1,3,4,6}
{1,2,5,6}

Using a brute force algorithm, I came up with the following maximums for width = {1, 2, 3, 4 ..., 24}

1, 1, 1, 1, 1, 2, 2, 3, 5, 8, 12, 20, 32, 58, 94, 169, 289, 526, 910, 1667, 2934, 5448, 9686, 18084

My algorithm breaks at 25 due to the huge memory trequirements needed to sample every possible combination. I plugged it into Sloan's, but no luck.

With a little tweaking, the series seems like it might have some sort of partial relationship with the Fibonacci and Lucas series, but I haven't been able to come up with anything concrete.

1, 1, 1, 1, 1, 2, 2, 3, 5, 8, 12, 20, 32, 58, 94, 169, 289, 526, 910
 ,  ,  ,  ,  ,  , 1, 2, 3, 5, 8,  13, 21, 34, 55,  89, 144, 233, 377 	(fib)
 --------------------------------------------------------------------
		  1, 1, 2, 3, 4,  7,  11, 24, 39,  86, 145, 293, 533	(partial lucas)
Anyone see the pattern? Or perhaps someone with Mathematica 7 could plug the series into the series calculator and come up with the formula?

Thanks for any help
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ramsey2879
#2
Jan24-10, 08:28 PM
P: 894
Quote Quote by ktoz View Post
Hi

I'm playing around with partitions and have come up with an integer sequence representing the maximum number of partitions of various "widths" that display the following properties:

- min values in partition are equal
- max values in partition are equal
- partitions contain equal number of members
- sum of members is equal

For example, given:

min = 1
max = 6
count = 4
sum = 14

There are only two partitions that satisfy the constraints

{1,3,4,6}
{1,2,5,6}

Using a brute force algorithm, I came up with the following maximums for width = {1, 2, 3, 4 ..., 24}

1, 1, 1, 1, 1, 2, 2, 3, 5, 8, 12, 20, 32, 58, 94, 169, 289, 526, 910, 1667, 2934, 5448, 9686, 18084

My algorithm breaks at 25 due to the huge memory trequirements needed to sample every possible combination. I plugged it into Sloan's, but no luck.

With a little tweaking, the series seems like it might have some sort of partial relationship with the Fibonacci and Lucas series, but I haven't been able to come up with anything concrete.

1, 1, 1, 1, 1, 2, 2, 3, 5, 8, 12, 20, 32, 58, 94, 169, 289, 526, 910
 ,  ,  ,  ,  ,  , 1, 2, 3, 5, 8,  13, 21, 34, 55,  89, 144, 233, 377 	(fib)
 --------------------------------------------------------------------
		  1, 1, 2, 3, 4,  7,  11, 24, 39,  86, 145, 293, 533	(partial lucas)
Anyone see the pattern? Or perhaps someone with Mathematica 7 could plug the series into the series calculator and come up with the formula?

Thanks for any help
Could you define "width"? You give 4 variables for widths but you give a sequence for widths = [1,2,3,...} so I don't understand what you mean.


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