Hexagonal Number
We consider dotted hexagons with 1, 2, 3, . . . dots on each side,
see also the picture. The number of dots in such a hexagon is
called a hexagonal number. The first hexagonal number is 1,
the second is 7, and the third is 19.
Which of the following numbers is also a hexagonal number?
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1 solution
I suppose they are going to move your question to "Discrete Math" section (against your will), but I post a solution regardless.
A recursive relation can be found for the sequence numbers.
H j = H j − 1 + 6 . ( j − 1 ) , H 0 = 1
then, you may use generating functions to find a closed form formula for the sequence.
H j = 1 + 3 j ( j + 1 )
In order to see which number is of such format, you can take them and subtract a unit form them. The result should be divisible by 3 . This way, only two choice remains 1 6 9 , 1 8 7 . Then you need to see which one is a multiplication of two consecutive integers, after 1 is subtracted and divided by 3 . 1 6 9 is thinly one that fits.