|
|
| Line 1: |
Line 1: |
| The '''Leibniz harmonic triangle''' is a [[triangle|triangular]] arrangement of [[fraction (mathematics)|fractions]] in which the outermost diagonals consist of the [[multiplicative inverse|reciprocal]]s of the row numbers and each inner cell is the [[absolute value]] of the cell above minus the cell to the left. To put it [[algebra]]ically, {{math|1=''L''(''r'', 1) = 1/''r''}} (where {{math|''r''}} is the number of the row, starting from 1, and {{math|1=''c''}} is the column number, never more than ''r'') and {{math|1=''L''(''r'', ''c'') = |''L''(''r'' - 1, ''c'' - 1) − ''L''(''r'', ''c'' - 1)|.}}
| | I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Office supervising is exactly where my main earnings arrives from but I've always wanted my personal business. What me and my family love is performing ballet but I've been taking on new things lately. Alaska is exactly where he's usually been residing.<br><br>Feel [http://www.weddingwall.com.au/groups/easy-advice-for-successful-personal-development-today/ free psychic readings] to [http://www.010-5260-5333.com/index.php?document_srl=1880&mid=board_ALMP66 psychic solutions by lynne] surf to my web blog: free tarot readings - [http://www.skullrocker.com/blogs/post/10991 simply click the up coming post], |
| | |
| The first eight rows are:
| |
| | |
| <math>\begin{array}{cccccccccccccccccc} | |
| & & & & & & & & & 1 & & & & & & & &\\
| |
| & & & & & & & & \frac{1}{2} & & \frac{1}{2} & & & & & & &\\
| |
| & & & & & & & \frac{1}{3} & & \frac{1}{6} & & \frac{1}{3} & & & & & &\\
| |
| & & & & & & \frac{1}{4} & & \frac{1}{12} & & \frac{1}{12} & & \frac{1}{4} & & & & &\\
| |
| & & & & & \frac{1}{5} & & \frac{1}{20} & & \frac{1}{30} & & \frac{1}{20} & & \frac{1}{5} & & & &\\
| |
| & & & & \frac{1}{6} & & \frac{1}{30} & & \frac{1}{60} & & \frac{1}{60} & & \frac{1}{30} & & \frac{1}{6} & & &\\
| |
| & & & \frac{1}{7} & & \frac{1}{42} & & \frac{1}{105} & & \frac{1}{140} & & \frac{1}{105} & & \frac{1}{42} & & \frac{1}{7} & &\\
| |
| & & \frac{1}{8} & & \frac{1}{56} & & \frac{1}{168} & & \frac{1}{280} & & \frac{1}{280} & & \frac{1}{168} & & \frac{1}{56} & & \frac{1}{8} &\\
| |
| & & & & &\vdots & & & & \vdots & & & & \vdots& & & & \\
| |
| \end{array}</math>
| |
| | |
| The denominators are listed in {{OEIS|id=A003506}}, while the numerators are all 1s.
| |
| | |
| Whereas each entry in [[Pascal's triangle]] is the sum of the two entries in the above row, each entry in the Leibniz triangle is the sum of the two entries in the row ''below'' it. For example, in the 5th row, the entry (1/30) is the sum of the two (1/60)s in the 6th row.
| |
| | |
| Just as Pascal's triangle can be computed by using binomial coefficients, so can Leibniz's: <math>L(r, c) = \frac{1}{r {r-1 \choose c-1}}</math>. Furthermore, the entries of this triangle can be computed from Pascal's, "the terms in each row are the initial term divided by the corresponding Pascal triangle entries."<ref>Wells, David (1986). ''The Penguin Dictionary of Curious and Interesting Numbers'', p.98. ISBN 978-0-14-026149-3.</ref>
| |
| | |
| This triangle can be used to obtain examples for the [[Erdős–Straus conjecture]] when ''n'' is divisible by 4.
| |
| | |
| If one takes the denominators of the ''n''th row and adds them, then the result will equal <math>n 2^{n - 1}</math>. For example, for the 3rd row, we have 3 + 6 + 3 = 12 = 3 × 2<sup>2</sup>.
| |
| | |
| It is worth noting that <math>L(r, c) = \int_0^1 \! x ^ {c - 1} (1 - x)^{r-c} \,dx \,.</math>
| |
| | |
| ==References== | |
| {{reflist}}
| |
| | |
| [[Category:Triangles of numbers]]
| |
| [[Category:Gottfried Leibniz]]
| |
| | |
| {{numtheory-stub}}
| |
I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Office supervising is exactly where my main earnings arrives from but I've always wanted my personal business. What me and my family love is performing ballet but I've been taking on new things lately. Alaska is exactly where he's usually been residing.
Feel free psychic readings to psychic solutions by lynne surf to my web blog: free tarot readings - simply click the up coming post,