perplexus.info :: Algorithms : Ceiling and Floor Formulation

Ceiling and Floor Formulation (Posted on 2013-03-04)

Formulate an algorithm for fast evaluation of:
Σ_j=1,...,n² (floor (√j) + ceil (√j)), where n is a positive integer.

** ceil(x) is the least integer ≥ x and, floor(x) is the greatest integer ≤ x

No Solution Yet Submitted by K Sengupta

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Solution

| Comment 5 of 6 |

Looking at the result for n = 4 we get:

ceil  1  2  2  2  3  3  3  3  3  4  4  4  4  4  4  4

j     1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16

floor 1  1  1  2  2  2  2  2  3  3  3  3  3  3  3  4

The ceilings produce 1x1, 3x2, 5x3, 7x4

The floors produce 3x1, 5x2, 7x3, 1x4

This suggests the answer can be obtained by the sum of squares of the first n odd numbers plus one plus n.

Posted by snark on 2013-03-05 20:12:00

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