Triangle patterns

Triangle patterns

In Pascal’s triangle, each number is the sum of the two numbers triangle patterns above it. In mathematics, Pascal’s triangle is a triangular array of the binomial coefficients.

0 and are usually staggered relative to the numbers in the adjacent rows. A diagram that shows Pascal’s triangle with rows 0 through 7. This recurrence for the binomial coefficients is known as Pascal’s rule. Pascal’s triangle has higher dimensional generalizations. The three-dimensional version is called Pascal’s pyramid or Pascal’s tetrahedron, while the general versions are called Pascal’s simplices. Yang Hui’s triangle, as depicted by the Chinese using rod numerals, appears in a mathematical work by Zhu Shijie, dated 1303. The pattern of numbers that forms Pascal’s triangle was known well before Pascal’s time.

Pingala in or before the 2nd century BC. In 1068, four columns of the first sixteen rows were given by the mathematician Bhattotpala, who was the first recorded mathematician to equate the additive and multiplicative formulas for these numbers. In the west, the binomial coefficients were calculated by Gersonides in the early 14th century, using the multiplicative formula for them. In this, Pascal collected several results then known about the triangle, and employed them to solve problems in probability theory. Pascal’s triangle determines the coefficients which arise in binomial expansions. Notice the coefficients are the numbers in row two of Pascal’s triangle: 1, 2, 1. An interesting consequence of the binomial theorem is obtained by setting both variables x and y equal to one.

In other words, the sum of the entries in the nth row of Pascal’s triangle is the nth power of 2. A second useful application of Pascal’s triangle is in the calculation of combinations. But this is also the formula for a cell of Pascal’s triangle. Rather than performing the calculation, one can simply look up the appropriate entry in the triangle.

Provided we have the first row and the first entry in a row numbered 0, the answer will be located at entry k in row n. For example, suppose a basketball team has 10 players and wants to know how many ways there are of selecting 8. By the central limit theorem, this distribution approaches the normal distribution as n increases. This is related to the operation of discrete convolution in two ways. Pascal’s triangle has many properties and contains many patterns of numbers.

Each frame represents a row in Pascal’s triangle. Each column of pixels is a number in binary with the least significant bit at the bottom. Light pixels represent ones and the dark pixels are zeroes. The sum of the elements of a single row is twice the sum of the row preceding it.

1, row 1 has a value of 2, row 2 has a value of 4, and so forth. This is because every item in a row produces two items in the next row: one left and one right. Pi can be found in Pascal’s triangle through the Nilakantha infinite series. Thus entries can simply be added in interpreting the value of a row.

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