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Does anyone know what's the general form of this sequence?

  1. Oct 5, 2011 #1

    KFC

    User Avatar

    Hi there,
    I am working on a problem about a combination of some items. The data set consists of three different sets. The first set are all combinations of six numbers from {0, 1, 2, 3, ..., 10, 11, 12}. The second set are all combinations of two numbers from {13, 14, 15, 16, ..., 24, 25}. The last set are all combinations of two numbers from {26, 27, 28, ..., 36, 37, 38}. The overall data set is a "product" of these three sets, i.e. all combinations of the data from first set, data from second set and data from last set.

    For the first glance, I thought the total number of line should be C(13,6)*C(13,2)*C(13,2) = 10440144, but actually, it only has 5286966 lines. The reason is the last data set is not completely independent of the second one. Namely, if we only consider the last set, all combination should be like

    26 27
    26 28
    26 29
    ...
    26 38
    27 28
    27 29
    27 30
    ...
    37 38

    If above data set is independent of the second set, after one cycle, above pattern will be repeated all over again. For example, the first two numbers of the middle set is
    13 14, when it proceed with increasing one step to 13 15, all combination of the last set will be repeated again. But I found that this is not the case, whenever the middle set proceed with one step, the last set will not be repeated from the very beginning. I list part of the data set below.

    We see that when the 13 14 increased to 13 15, the last set does not repeat from 26 27, instead it starts from 26 28.

    If we trace more line, we found that when the middle set changed to 13 16, the last set start its cycle from 26 29.

    When the middle set changed to 14 15, the last set start its cycle from 27 28.

    Does any one know in what way to generate this data? Any formula to help generating data of these patterns? What's the application of this pattern?

    0 1 2 3 4 5 13 14 26 27
    0 1 2 3 4 5 13 14 26 28
    0 1 2 3 4 5 13 14 26 29
    0 1 2 3 4 5 13 14 26 30
    0 1 2 3 4 5 13 14 26 31
    0 1 2 3 4 5 13 14 26 32
    0 1 2 3 4 5 13 14 26 33
    0 1 2 3 4 5 13 14 26 34
    0 1 2 3 4 5 13 14 26 35
    0 1 2 3 4 5 13 14 26 36
    0 1 2 3 4 5 13 14 26 37
    0 1 2 3 4 5 13 14 26 38
    0 1 2 3 4 5 13 14 27 28
    0 1 2 3 4 5 13 14 27 29
    0 1 2 3 4 5 13 14 27 30
    0 1 2 3 4 5 13 14 27 31
    0 1 2 3 4 5 13 14 27 32
    0 1 2 3 4 5 13 14 27 33
    0 1 2 3 4 5 13 14 27 34
    0 1 2 3 4 5 13 14 27 35
    0 1 2 3 4 5 13 14 27 36
    0 1 2 3 4 5 13 14 27 37
    0 1 2 3 4 5 13 14 27 38
    0 1 2 3 4 5 13 14 28 29
    0 1 2 3 4 5 13 14 28 30
    0 1 2 3 4 5 13 14 28 31
    0 1 2 3 4 5 13 14 28 32
    0 1 2 3 4 5 13 14 28 33
    0 1 2 3 4 5 13 14 28 34
    0 1 2 3 4 5 13 14 28 35
    0 1 2 3 4 5 13 14 28 36
    0 1 2 3 4 5 13 14 28 37
    0 1 2 3 4 5 13 14 28 38
    0 1 2 3 4 5 13 14 29 30
    0 1 2 3 4 5 13 14 29 31
    0 1 2 3 4 5 13 14 29 32
    0 1 2 3 4 5 13 14 29 33
    0 1 2 3 4 5 13 14 29 34
    0 1 2 3 4 5 13 14 29 35
    0 1 2 3 4 5 13 14 29 36
    0 1 2 3 4 5 13 14 29 37
    0 1 2 3 4 5 13 14 29 38
    0 1 2 3 4 5 13 14 30 31
    0 1 2 3 4 5 13 14 30 32
    0 1 2 3 4 5 13 14 30 33
    0 1 2 3 4 5 13 14 30 34
    0 1 2 3 4 5 13 14 30 35
    0 1 2 3 4 5 13 14 30 36
    0 1 2 3 4 5 13 14 30 37
    0 1 2 3 4 5 13 14 30 38
    0 1 2 3 4 5 13 14 31 32
    0 1 2 3 4 5 13 14 31 33
    0 1 2 3 4 5 13 14 31 34
    0 1 2 3 4 5 13 14 31 35
    0 1 2 3 4 5 13 14 31 36
    0 1 2 3 4 5 13 14 31 37
    0 1 2 3 4 5 13 14 31 38
    0 1 2 3 4 5 13 14 32 33
    0 1 2 3 4 5 13 14 32 34
    0 1 2 3 4 5 13 14 32 35
    0 1 2 3 4 5 13 14 32 36
    0 1 2 3 4 5 13 14 32 37
    0 1 2 3 4 5 13 14 32 38
    0 1 2 3 4 5 13 14 33 34
    0 1 2 3 4 5 13 14 33 35
    0 1 2 3 4 5 13 14 33 36
    0 1 2 3 4 5 13 14 33 37
    0 1 2 3 4 5 13 14 33 38
    0 1 2 3 4 5 13 14 34 35
    0 1 2 3 4 5 13 14 34 36
    0 1 2 3 4 5 13 14 34 37
    0 1 2 3 4 5 13 14 34 38
    0 1 2 3 4 5 13 14 35 36
    0 1 2 3 4 5 13 14 35 37
    0 1 2 3 4 5 13 14 35 38
    0 1 2 3 4 5 13 14 36 37
    0 1 2 3 4 5 13 14 36 38
    0 1 2 3 4 5 13 14 37 38
    0 1 2 3 4 5 13 15 26 28
    0 1 2 3 4 5 13 15 26 29
    0 1 2 3 4 5 13 15 26 30
    0 1 2 3 4 5 13 15 26 31
    0 1 2 3 4 5 13 15 26 32
    0 1 2 3 4 5 13 15 26 33
    0 1 2 3 4 5 13 15 26 34
     
  2. jcsd
  3. Oct 5, 2011 #2

    KFC

    User Avatar

    I find some clue regarding the pattern of the data set. Let's denote all combination of the middle set and the last set (i.e. last 4 numbers) as

    A B C D

    now consider the modulus on the last two numbers

    A1 = C%13
    B1 = D%13

    and consider the mapping

    C1 = A + 13
    D1 = B + 13

    now only for those combinations such that no more than one sequence A1 B1 C1 D1 found in A B C D. But I still don't found the general formula to help for listing all the cases
     
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