2005

[Marc LeBrun]

Happy MMV!

 From the OEIS we learn, among many other things, that 2005 is...

   Vertically symmetric  [A053701]

   6 4^5 + 5 4^4 + 4 4^3 + 3 4^2 + 2 4^1 + 1 4^0  [A059045]

   The value of 11 n^2 + 11 n + 3 for n=13  [A006222]

   The value of (14 n^3  - 21 n^2 + 13 n)/4 for n=10  [A071229]

   The value of ((2n - 1) 3^n + 1)/4 for n=6  [A014915]

   The coefficient of x^14 in (1-x)/(1-x-x^2-x^3-x^4+x^5)  [A033305]

   Pick integers x, y and z between 1 and 32 inclusive.  The expression 
xy+yz+zx can take on any one of exactly 2005 distinct values   [A100440, 
recently contributed on 21 Nov 04].

   Expand (p+1)^8-1 and then "umbrally" replace p^n by the n-th prime.  The 
result sums to 2005  [A050513]

   Perhaps the most interesting and apropos is the following variant of the 
well-known "birthday paradox": Pick 53 years at random since the birth of 
Christ (ie any AD) and the chances are just better than half that two will 
coincide, but pick only 52 and the odds are just less than even.  [A072829, 
if I've interpreted it aright]

Hope you find these factoids of interest.  Best wishes for the New Year!
--Marc LeBrun