[seqfan] Re: A002251 rediscovered as "home-sick queens"

wouter meeussen wouter.meeussen at pandora.be
Fri Nov 4 13:07:07 CET 2011


a small handicap on those queens:
they attack left and right and up and down, but diagonaly only right-down and up-left,
but *not* diagonaly right-up and left-down.
It's those handicaped queens that generate A002251.

If we add the second (anti-)diagonal, using "UnsameQ@@ Plus@@@ #"
as the fourth backtrack condition, then we find Antti's A065188 :
"Greedy Queens" permutation of the natural numbers, inverse of A065189.
with offset 0 instead of 1.

The handicaped queens take positions below the main diagonal for
(2,1)  (5,3)  (7,4) (10,6) ...
 on index
3,6,8,11,14,16,19,21   also known as A026352 :
[ n*tau ]+n+1.  (* read as Floor[k*GoldenRatio]+k+1 *)

but the standard queens (attacking along both diagonals) take positions
below the main diagonal on index
4, 5, 10, 11, 13, 15, 20, 21, 26, 27, 29, 31, 34, 38, 40
and that is not in OEIS and superseeker came up blank.
Would it be of interest to add it?

Wouter.


  ----- Original Message ----- 
  From: wouter meeussen 
  To: SeqFan 
  Sent: Wednesday, November 02, 2011 11:09 AM
  Subject: A002251 rediscovered as "home-sick queens"


  A graphical analogue is sometimes nice to explain an otherwise dull (?) sequence.

  On a large chessboard, place a queen in the upper-left corner at (0,0);
  then add non-attacking queens as close as possible to the origin.
  You get
  {0, 0}, {1, 2}, {2, 1}, {3, 5}, {4, 7}, {5, 3}, {6, 10}, {7, 4}, {8, 13}, {9, 15},
   {10, 6}, {11, 18}, {12, 20}, {13, 8}, {14, 23}, {15, 9}, {16, 26}, {17, 28}, {18, 11} etc
  The row-index is simply increasing,
  the column index is A002251:
  "Start with sequence of nonnegative integers; then swap L(k) and U(k) for all k >= 1, 
  where L = A000201, U = A001950 (lower and upper Wythoff sequences)."

  Isn't this correct?

  Wouter.

  inspired by http://www.physicsforums.com/showthread.php?p=3590295&posted=1#post3590295
  ----------------------------------------------------
  Mma 4.0 implementation (lazy programmer's)
  In[1]:=<<DiscreteMath`Combinatorica`
  In[2]:=base=Outer[List,Range[0,64],Range[0,2*64]];
  In[3]:=Backtrack[base,(And[UnsameQ@@First /@#,UnsameQ@@Last/@#,UnsameQ@@Subtract@@@#])&,True&,One]
  -----------------------------------------------------


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