Sequence

[Whateverist]

I'm no mathamatision. But, I like to entertain myself with numbers. My latest toy is a sequence of odd numbers.

If I have one square tile and I want to increase it's size, keeping it square with the same size tiles, I would have to add 3 tiles.

Then I would then have 4 tiles. If I wanted to increase it's size from there, I would have to add 5 tiles for a total of 9 tiles at level 3. Increasing in size then is a sequence: 1 3 5 7 9 11 13... The 12th level is 144 tiles: 10th level = 100 tiles + 21 tiles = 121 tiles(11th level) + 23 tiles = 144.

Each level of tiles, 1 - 12, is the square root of the total number of tiles at each level. I think it's cool that this sequence at the 12th level is the same number as the 12th level of the Fibonacci sequence.

Anyway, I thought is was fun. What does it mean? I have no idea.

Checking:

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Fibonacci numbers: 1, 1, 2, 3, 5,    8, 13, 21, 34, 55,   89, 144  Yup, the 12th Fibonacci number is the same as the number of tiles in the 12th N by N square.

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 This is also true for first fibonacci number and the first square number.  Will it ever be true again?

Also, the number of tiles that must be added to get to the Nth level square from the one before it is

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2N – 1

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So to get to the 2nd, two by two, square from the one before it  (1 by 1)  you must add 3 tiles.  To get from to 33rd square from the one before it, you'd need to add 65 squares.  How many tiles would you need to add to get to the 1,003rd square?  

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2,005, which will be the 1,003rd odd number starting with 1, of course.

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