Sum Of Multiples Of Sqrt 2 Floored

Doctor vogler unpacks it enough to convince himself that the sum might lack a closed form and so sets about deriving an approximation.

Sum of multiples of sqrt 2 floored. About sum summation calculator. Square root of sum versus sum of square roots 01 sep 2016 21 58 display posts from previous. Look at the difference f 1 2 n s n. I would like to prove rigorously not intuitively that sum n 1 n n sqrt 2 frac n 2 mathcal o sqrt n where is the fractional part function.

The sum summation calculator is used to calculate the total summation of any set of numbers. Stack exchange network consists of 176 q a communities including stack overflow the largest most trusted online community for developers to learn share their knowledge and build their careers. An adult seeks a closed form solution for a complicated sum. Since n can be very large up to 101 digits using just sqrt 2 and a loop won t work.

Now it s a known result that frac nr is uniformly distributed in an open interval 0 1 and therefore for big n is close to n 2. Hereupon r sqrt 2. Sometimes it s easier to take a step back and concentrate not on what you have in front of you but on what you. It also means that the number of integer triangles with even numbered perimeters p 2n is the same as the number of integer triangles with odd numbered perimeters p 2n 3.

For example if str n was 5 the answer would be calculated as floor 1 sqrt 2 floor 2 sqrt 2 floor 3 sqrt 2 floor 4 sqrt 2 floor 5 sqrt 2 1 2 4 5 7 19 so the function would return 19. Actually there are slightly more than 1m digits here. This is the integer closest to p 2 48 when p is even and to p 3 2 48 when p is odd. I understand intuitively why this is true and that s how i came up with this claim n sqrt 2 behaves like a random variable uniformly distributed in 0 1 and treating it as a random variable makes the sum.

So s n is very close to f 1 2 n n 2. That is for every number i in the range 1 to n it adds up all of the integer portions of i sqrt 2. Floor 5 sqrt 2 1 2 4 5 7 19 so the function would return 19. All posts 1 day 7 days 2 weeks 1 month 3 months 6 months 1 year sort by author post time subject ascending descending.

Str n will be a positive integer between 1 and 10 100 inclusive. The result is their sum or total. These digits were computed by robert nemiroff george mason university and nasa goddard space flight center and checked by jerry bonnell university space research association and. It s easy to see it equals to the integer part of the sum frac r frac 2r frac nr.