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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.

Anyone see the pattern? Or perhaps someone with Mathematica 7 could plug the series into the series calculator and come up with the formula?Code (Text):

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)

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1, 1, 2, 3, 4, 7, 11, 24, 39, 86, 145, 293, 533 (partial lucas)

Thanks for any help

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# Help deriving integer sequence formula

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