[HN Gopher] Pi in Pascal's Triangle (2014)
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       Pi in Pascal's Triangle (2014)
        
       Author : senfiaj
       Score  : 41 points
       Date   : 2025-06-06 12:38 UTC (3 days ago)
        
 (HTM) web link (www.cut-the-knot.org)
 (TXT) w3m dump (www.cut-the-knot.org)
        
       | lixtra wrote:
       | (2014) [1]
       | 
       | [1]
       | https://web.archive.org/web/20140901000000*/https://www.cut-...
        
       | zkmon wrote:
       | I wonder if the triangle hides any secrets related to prime
       | numbers as well.
        
         | dhosek wrote:
         | It's possible, I suppose.
         | 
         | One of my favorite proofs that the sums of each row are powers
         | of two comes from the fact that the numbers in row n+1 are the
         | coefficients of the powers of (a+b)n, so setting a=b=1 you get
         | 2n (most discrete math students seeking to prove this end up
         | reaching for induction which is a heavier proof than this).
        
       | JohnKemeny wrote:
       | p - 2 = 1/1 + 1/3 - 1/6 - 1/10 + 1/15 + 1/21 - 1/28 ***
       | 
       | That's beautiful. I wonder why the -2 is there, though. To fix
       | it, we would need
       | 
       | p = -x + 1/1 + 1/3 - 1/6 - 1/10 + 1/15 + 1/21 - 1/28 ***
       | 
       | where x = 2, and so it would be
       | 
       | p = -1/1/2 + 1/1 + 1/3 - 1/6 - 1/10 + 1/15 + 1/21 - 1/28 ***
       | 
       | which makes the -1st triangle number 1/2, I guess.
        
         | dhosek wrote:
         | 0th triangle number and I think that can probably be connected
         | to 1+1-1+1-1+[?]=1/2 somehow.
        
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