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Thread: Number stairs

  1. #1
    4 Star Lounger
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    Number stairs

    In the number grid below, there can be placed stair shapes as in the two examples - turquoise represents a 4 by 4 stair, the tan a 3 by 3 stair.
    Let N represent the bottom left number of a stair shape.

    It can be shown that the sum of the numbers in a stair shape is given by:
    3 by 3.......6N+44
    4 by 4.......10N+110
    5 by 5.......15N+220
    6 by 6.......21N+385
    7 by 7.......28N+616
    8 by 8.......36N+924
    9 by 9.......45N+1320
    10 by 10...55N+1815

    It is easy to see the relationship between the first number in each row and the next, but what is the relationship between successive second numbers?
    <font color=blue><font face="Script MT Bold"><big>Rob</big></font face=script></font color=blue>

  2. #2
    Plutonium Lounger
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    Re: Number stairs

    <span style="background-color: #FFFF00; color: #000000; font-weight: bold"><font color=yellow>For an s by s stair, the second number is 11/6*(s-1)*s*(s+1).
    If you calculate the differences between the numbers, you get 66, 110, 165, 231, ...
    If you calculate the differences of that sequence, you get 44, 55, 66, ...
    If you calculate the differences of that sequence, you get 11, 11, ...
    Since the 3rd order differences are constant, the original sequence is a 3rd degree polynomial.</font color=yellow></span hi>

  3. #3
    4 Star Lounger
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    Re: Number stairs

    Bravo, Hans!
    <font color=blue><font face="Script MT Bold"><big>Rob</big></font face=script></font color=blue>

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