Lacsap's Triangle
1
Introduction.
Let us consider a triangle of fractions:
Obviously, the numbers are following some pattern. In this investigation we will try to explain the theory behind this arrangement and to find a general relation between the element’s number and its value.
The pattern above is called a Lacsap’s Triangle, which inevitably hints at its relation to another arrangement - Pascal’s Triangle (as Lacsap appears to be an anagram of
Pascal).
The algorithm behind it is very simple: each element is the sum of the two elements above it. However, if we represent a triangle as a table (below), we will be able to notice a pattern between an index number of an element and its value: column column
column
column
column
column
column
2
0
1
2
3
4
5
row 0
1
row 1
1
1
row 2
1
2
1
row 3
1
3
3
1
row 4
1
4
6
4
1
row 5
1
5
10
10
5
1
row 6
1
6
15
20
15
6
6
1
It seems important to us to stress several points that this table makes obvious:
● the number of elements in a row is n + 1 (where n is an index number of a row)
● the element in column 1 is always equal to the element in column n - 1
● therefore, the element in column 1 in every row is equal to the number of a given row. Now when we have established the main sequences of a Pascal’s triangle let us see how they are going to be expressed in a Lacsap’s arrangement. We also suggest looking at numerators and denominators separately, because it seems obvious that the fractions themselves can’t be derived from earlier values using the progressions of the sort that Pascal uses.
Finding Numerators.
Let’s begin with presenting given numerators in a similar table, where n is a number of a row. n=1
1
1
n=2
1
3
1
n=3
1
6
6
1
n= 4
1
10
10
10
1
n=5
1
15
15
15
15
1
3
Although the triangles
Introduction.
Let us consider a triangle of fractions:
Obviously, the numbers are following some pattern. In this investigation we will try to explain the theory behind this arrangement and to find a general relation between the element’s number and its value.
The pattern above is called a Lacsap’s Triangle, which inevitably hints at its relation to another arrangement - Pascal’s Triangle (as Lacsap appears to be an anagram of
Pascal).
The algorithm behind it is very simple: each element is the sum of the two elements above it. However, if we represent a triangle as a table (below), we will be able to notice a pattern between an index number of an element and its value: column column
column
column
column
column
column
2
0
1
2
3
4
5
row 0
1
row 1
1
1
row 2
1
2
1
row 3
1
3
3
1
row 4
1
4
6
4
1
row 5
1
5
10
10
5
1
row 6
1
6
15
20
15
6
6
1
It seems important to us to stress several points that this table makes obvious:
● the number of elements in a row is n + 1 (where n is an index number of a row)
● the element in column 1 is always equal to the element in column n - 1
● therefore, the element in column 1 in every row is equal to the number of a given row. Now when we have established the main sequences of a Pascal’s triangle let us see how they are going to be expressed in a Lacsap’s arrangement. We also suggest looking at numerators and denominators separately, because it seems obvious that the fractions themselves can’t be derived from earlier values using the progressions of the sort that Pascal uses.
Finding Numerators.
Let’s begin with presenting given numerators in a similar table, where n is a number of a row. n=1
1
1
n=2
1
3
1
n=3
1
6
6
1
n= 4
1
10
10
10
1
n=5
1
15
15
15
15
1
3
Although the triangles
Bibliography: 1) Weisstein, Eric W. "Pascal 's Triangle." From MathWorld--A Wolfram Web Resource. http:// mathworld.wolfram.com/PascalsTriangle.html 2) “Pascal’s Triangle and Its Patterns”; an article from All you ever wanted to know http:// ptri1.tripod.com/ 3) Lando, Sergei K.. "7.4 Multiplicative sequences". Lectures on generating functions. AMS. ISBN 0-8218-3481-9