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Topic: Pascal's triangle (Read 2767 times) |
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Christine
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Posts: 159
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Pascal's triangle
« on: Sep 9th, 2013, 11:32am » |
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A073617 a(n) = product of terms at +45 degrees slope with the horizontal.
1, 1, 1, 2, 3, 12, 30, 240, 1050, 16800, 132300, 4233600, 61122600, 3911846400, 104886381600, 13425456844800, 674943865596000, 172785629592576000, 16407885372638760000, 8400837310791045120000,
1515727634953623371280000
http://oeis.org/A073617
Taking sets of 4 consecutive terms, then
the product of the 1-st and 4-th divided by the product of the 2-nd and 3-rd
1, 1, 1, 2
(1*2)/(1*1) = 2
1, 1, 2, 3
(1*3)/(1*2) = 3/2 = 1.5
1, 2, 3, 12
(1*12)/(2*3) = 2
2, 3, 12, 30
(2*30)/(3*12) = 5/3 = 1.666..
3, 12, 30, 240
(3*240)/(12*30) = 2
when starting with an even indexed row the ratio is 2
Will the ratio always be 2?
with odd indexed row, the ratios are
1.5
1.666..
1.75
1.8
1.83333...
1.857142...
1.875
1.88888...
1.90
Will the ratio reach 2 ?
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towr
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Re: Pascal's triangle
« Reply #1 on: Sep 12th, 2013, 1:02am » |
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on Sep 9th, 2013, 11:32am, Christine wrote:| when starting with an even indexed row the ratio is 2 |
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WolframAlpha confirms that that's the case. I haven't managed to do the calculation by hand though. And the odd case is little more complex than wolframalpha is willing to do without further simplification.
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Wikipedia, Google, Mathworld, Integer sequence DB
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