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Leibniz and Pascal Triangles

The applet below presents Pascal and Leibniz triangles modulo a specified number. (In addition, all

entries exceeding 10 are displayed modulo 10. Also, for the Leibniz triangle whose entries are unit

fractions, i.e. fraction with 1 in the numerator, we use their whole reciprocals.)

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The Pascal Triangle is a standard fixture in recreational mathematics. Formed by binomial

coefficients it possesses an inordinate number of interesting properties (see the references below.)

Its construction starts from the top. In every row, the first and the last numbers equal 1. A generic

entry is obtained by summing up the two entries just above it.

The Leibniz Triangle [Polya, p 88], also called the Leibniz Harmonic Triangle, is by far less known,

although it relates to that of Pascal in a very simple way.

A row of the Leibniz triangle starts with the reciprocal of the row number (or the row number plus

one depending on whether one starts counting from 1 or 0.) Every entry is the sum of the two

numbers just below it. The entries can thus be computed sequentially left to right and top to bottom

using subtraction instead of addition. E.g., in the fifth row, 1/20 = 1/4 - 1/5, 1/30 = 1/12 - 1/20,

1/20 = 1/12 - 1/30, 1/5 = 1/4 - 1/20. Asymmetry of construction notwithstanding, the triangle is

symmetric with respect to its vertical axis.

The sequence of the second entries: 1/2, 1/6, 1/12, 1/20, ... forms the telescoping series:

1/(1·2) + 1/(2·3) + 1/(3·4) + 1/(4·5) + ... =

(1/1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + (1/4 - 1/5) + ...,

in which all the terms eventually cancel out except for the very first, thus giving the sum of 1.

(There is a geometric illustration for the telescoping series.) It appears that the first term in the

first row is the sum of the second terms from all rows starting with the second one. This property is

generalized: the last term in the nth row is the sum of the (n + 1)-st terms from all the rows starting

with the (n + 1)-st. E.g.,

1/2 = 1/3 + 1/12 + 1/30 + 1/60 + ...

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Leibniz and Pascal Triangles http://www.cut-the-knot.org/Curriculum/Combinatorics/LeibnitzTriangle...

1 od 3 24.3.2013 15:25

1/3 = 1/4 + 1/20 + 1/60 + 1/140 + ...

1/4 = 1/5 + 1/30 + 1/105 + ...

mth element in the nth row of the Pascal triangle equals the binomial coefficient C(n, m), where we

start counting from 0. The corresponding entry of the Leibniz triangle is the reciprocal of

(n + 1)C(n, m). (In particular, it can be verified that the reciprocals of (n + 1)C(n, m) and

(n + 1)C(n, m + 1) add up to the reciprocal of nC(n - 1, m).)

References

R. Graham, D. Knuth, O. Patashnik, Concrete Mathematics, 2nd edition, Addison-Wesley, 19941.

H. E. Huntley, The Divine Proportion, Dover, 19702.

J. A. Paulos, Beyond Numeracy, Vintage Books, 19923.

C. A. Pickover, Computers, Patterns, Chaos, and Beauty, St. Martin's Press, 19904.

G. Polya, Mathematical Discovery, Wiley, 19815.

I. Stewart, Game, Set and Math, Penguin Books, 19896.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 19877.

D. Wells, You Are a Mathematician, Wiley, 19958.

Related material

Read more...

Infinite Sums and Products

Sum of an infinite series

Harmonic Series And Its Parts

A Telescoping Series

An Inequality With an Infinite Series

That Divergent Harmonic Series

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CITE THIS PAGE AS:A. Bogomolny, Leibniz and Pascal Triangles from Interactive Mathematics Miscellany and Puzzleshttp://www.cut-the-knot.org/Curriculum/Combinatorics/LeibnitzTriangle.shtml, Accessed 24 March 2013

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Leibniz and Pascal Triangles http://www.cut-the-knot.org/Curriculum/Combinatorics/LeibnitzTriangle...

2 od 3 24.3.2013 15:25