Sierpinski Triangle Adjacency

Probability Level 5

For each positive step integer $n$ , a set of triangles consists of numbers, which indicate how many triangles inside are either vertex- or edge-adjacent to another. The following is the explanation of the diagram above:

If $n = 1$ , then we have only one triangle, so the value is $1$ .

If $n = 2$ , then we count four mini triangles. Since each triangle is tangent to the other three, their values are $4$ .

If $n = 3$ , then we count thirteen triangles inside. Then, each set of smaller triangles contains different values position-wise.

Define $A(n)$ to be the sum of all triangle values for each $n$ . Considering the diagram and the pattern, evaluate $A(20)$ .

The answer is 15684238348.

1 solution

Christopher Criscitiello
Oct 20, 2017

The recursive formula is

$A_n = 3*(A_{n-1} + 2 + (2^{n-1} - 1))+3(2^{n-1}-1)+1, A_1=1$

which has explicit form

$A_n = 0.5*(-1 - 3*2^{2 + n} + 3^{2 + n})$

Got a proof?

Pi Han Goh - 3 years, 7 months ago

Sierpinski Triangle Adjacency

The answer is 15684238348.

1 solution

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