the two directly above it [ fig 6 ]. This triangle has many different uses and patterns, since the ( + 1) th row conveys the coefficients of a binomial expansion of the n th order.
fig. 7: Pascal’s Triangle with even numbers in white and odds in purple
fig. 6: Pascal’s Triangle
Here we see an interesting pattern if we colour each even number one colour and each odd number another. This colouring rule gives rise to Sierpinski’s Triangle [ fig. 7 ]. While this pattern is interesting for odd and even numbers, i.e. division by two, similar patterns emerge from divisions by other integers, each producing a slightly different pattern. However, in Pascal’s Triangle the numbers divisible by all of the prime numbers greater than or equal to 5 produce the same pattern but at different scales, [ fig. 8 ].
Division by 3
Division by 6
Division by 7
Division by 11
fig. 8: Pascal’s Triangle divided through by other integers
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