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Math Help - Shape/number sequences in different dimensions.

  1. #1
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    Shape/number sequences in different dimensions.

    I have recently been investigating how number sequences relate to each other when put into different dimensions. The simplest example is squares, where the Nth term is T(n)=n^d (d=number of dimensions). So for a square (in two dimensions), the pattern is T(n)=n^2, so we get 1,4,9,16 etc. Then, when in three dimensions, the pattern is T(n)= n^3, so we get 1,8,27,64 etc. This obviously can go on for an infinite number of dimensions.
    I then decided to try equilateral triangles. For triangular numbers, I worked out the pattern as T(n)=(n(n+1))/2. Then for tetrahedral numbers, I found that the pattern was T(n)=(n(n+1)(n+2))/6. Now this looks to me like if you add a dimension, you multiply the previous Nth term rule by (n+d-1)/d. However, I have no idea how to write this whole calculation! The furthest I can get is (n...whatever I'm missing...)/d! . Please Help!
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  2. #2
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    Hello, ToTheWareMobile!

    I have recently been investigating how number sequences relate to each other when
    put into different dimensions. The simplest example is squares, where the Nth term is
    T(n)=n^d (d=number of dimensions). So for a square (in two dimensions), the
    pattern is T(n)=n^2, so we get 1,4,9,16 etc. Then, when in three dimensions,
    the pattern is T(n)= n^3, so we get 1,8,27,64 etc.
    This obviously can go on for an infinite number of dimensions.

    I then decided to try equilateral triangles.
    For triangular numbers, I worked out the pattern as T(n)=(n(n+1))/2.
    Then for tetrahedral numbers, I found that the pattern was T(n)=(n(n+1)(n+2))/6.

    Now this looks to me like if you add a dimension, you multiply the previous Nth term
    rule by (n+d-1)/d. However, I have no idea how to write this whole calculation!
    The furthest I can get is (n...whatever I'm missing...)/d!
    Please help!

    I don't think I can help with The Big Picture but . . .


    Many years ago, I played with multi-deimensional "triangular" numbers.

    It was this time of year and a few classmates and I were drinking eggnog
    . . and singing "The Twelve Days of Christmas".

    We already knew about Triangular Numbers but someone asked,
    . . "How many gifts were given during The Twelve Days?"

    A little scribbling and we derived the formula for Tetrahedral Numbes
    . . (a term we thought we invented): . \dfrac{n(n+1)(n+2)}{3!}


    We had more eggnog and wondered about the sum of Tetrahedral numbers.
    . . This turned out to be: . \dfrac{n(n+1)(n+2)(n+3)}{4!}

    We called them "Pentahedroidal" numbers.
    In 4-space, this figure would be bounded by five tetrahedrons,
    . . and would be called a pentahedroid.


    We also wondered what scenario would produce this problem.

    Suppose My True Love decided to spread the gift-giving over twelve years.

    . . The first year, she gives me \,1 gift.

    . . The second year, she gives me 1 + 2 gifts.

    . . The third year,  1 + 2 + 3 gifts.

    . . . . . . . . . . . . . \vdots

    . . The twelfth year, 1 + 2 + 3 + \hdots + 12 gifts.



    At this point we ran out of eggnog and brain cells . . .

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