Christmas tree sequence...and a happy new Year !
Here are the first steps of a "Christmas tree" sequence :
How many triangular tiles are there in the th tree?
NB : the star is made of triangular tiles.
The answer is 2018.
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1 solution
Appropriately, the answer is 2 0 1 8 .
Three n consists of n layers of three rows each. Number these layers (from top to bottom) k = 1 , 2 , … , n .
Layer k contains ( 2 k + 1 ) + ( 2 k + 3 ) + ( 2 k + 5 ) = 6 k + 9 triangles.
Thus tree n contains (including the two triangles for the star) T ( n ) = 2 + k = 1 ∑ n 6 k + 9 = 2 + 6 ⋅ 2 1 n ( n + 1 ) + 9 n = 3 n 2 + 1 2 n + 2 triangles . Now substitute n = 2 4 .