The full paper develops the necessary tools to analyse the interesting sequences in more detail and also summarises the motivating properties of Pascal's Triangle
, including its fractal-like structure: colouring the odds and evens black and white results in the famous Sierpinski Triangle, if zoomed out far enough.
We investigate infinite log-concavity of the columns and other lines of Pascal's triangle
in Section 4.
Proof without words: Refer to the following Pascal's Triangle
(2) and see the underlined numbers.
This definition included a binomial term, and the mathematics simplified to the point where the second number in the given row of Pascal's Triangle
turned out to play an important role (see Figure 3).
So, having reeled off all sorts of numbers after applying Pascal's Triangle
and Poisson's Distribution (whatever they are), what has he come up with?
If the resulting signal from each overflow is tracked through the chain of summers, delay (D) and inverter (I) circuits, the same Pascal's triangle
sequence as that disclosed in Wells is built up.
The numbers 22, 343, 4664, 58985, 613316, 7367637, and 85922958, for example, can be identified in the Vedic matrix and arranged in Pascal's triangle
. Another interesting pattern shows that excluding 3, 6, and 9, the matrix contains six 1's and six 8s, six 2s and six 7s, and six 4s and six 5s.
A senior at the North Carolina School of Science and Mathematics in Durham, Reither won a $40,000 scholarship for finding the dimensions of fractals generated by Pascal's triangle
and its higher analogs.
Newcastle University team captain Jonathan Noble, left, and teammate Adam Lowery smile after Noble correctly identified the seventh line of Pascal's Triangle
Schiller and Charles (2004) conclude their article Moving Forward and Backward with Palindromes by asking how many times you can multiply 11 and produce a row of Pascal's triangle
. The first four powers of 11 produce palindromes that may interest middle school students whose teacher introduces them to this peculiarity.
She noticed that outcomes were triangular numbers, so she reasoned that perhaps she could find a connection to Pascal's triangle
. The teacher noticed that the sequence formed from the handshake problem could be found in the third position of every row, beginning with row 2 (fig.
This problem lays a foundation for a revisit when considering Pascal's Triangle
in later years.