Pascal’s Triangle and the Pell Family

Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.2. From Pascal to LeibnizIn Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.Specifically, we define the sequence sn; as follows [6]:

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3057. A Note on Pascal's Triangle

The Mathematical Gazette ◽

10.2307/3612055 ◽

1963 ◽

Vol 47 (359) ◽

pp. 57

Author(s):

Robert Croasdale

Keyword(s):

Pascal’S Triangle ◽

Pascal's Triangle

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The Fibonacci p-numbers and Pascal’s triangle

Cogent Mathematics ◽

10.1080/23311835.2016.1264176 ◽

2016 ◽

Vol 3 (1) ◽

pp. 1264176 ◽

Cited By ~ 1

Author(s):

Kantaphon Kuhapatanakul ◽

Lishan Liu

Keyword(s):

Pascal’S Triangle ◽

Pascal's Triangle

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Arithmetic Triangle

Journal of Mathematics Research ◽

10.5539/jmr.v9n2p100 ◽

2017 ◽

Vol 9 (2) ◽

pp. 100

Author(s):

Luis Dias Ferreira

Keyword(s):

Arithmetic Progression ◽

Modus Operandi ◽

Interesting Point ◽

Pascal’S Triangle ◽

Initial Term ◽

The Common ◽

Pascal's Triangle

The product of the first $n$ terms of an arithmetic progression may be developed in a polynomial of $n$ terms. Each one of them presents a coefficient $C_{nk}$ that is independent from the initial term and the common difference of the progression. The most interesting point is that one may construct an "Arithmetic Triangle'', displaying these coefficients, in a similar way one does with Pascal's Triangle. Moreover, some remarkable properties, mainly concerning factorials, characterize the Triangle. Other related `triangles' -- eventually treated as matrices -- also display curious facts, in their linear \emph{modus operandi}, such as successive "descendances''.

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