[seqfan] Re: Need help finding prime index values for sequence A363381

[martin_n_fuller at btinternet.com]

I have found 7 solutions for a(9) based on 1-dimensional tilings.  I haven't
found any other solutions but the search is incomplete.

Here are the 1-dimensional tiles that can tile a discrete interval of length
9 with reflections allowed:
  X ; XXX ; XX.X ; X..X..X ; XXXXXXXXX

NB the middle tile requires reflections:
  XX___X___
  ___XX___X
  __X___XX_

Any cartesian product of 2 of these tilings can tile a 9 x 9 square.  7 of
these products contain 9 elements so are solutions for A363381.
For example, the middle tiling multiplied by itself gives the following.
This can tile a 9 x 9 square using 4 copies in this orientation, 2 copies
reflected vertically, 2 copies reflected horizontally, and 1 copy rotated
180 degrees.
XX___X
XX___X
______
______
______
XX___X

The same technique works for even n, e.g. see page 98 of "The 102 6-cell
patterns for a 6 X 6 square".  But there are other solutions not based on
1-dimensional tilings.  I particularly like pages 11, 72 and 87. 
For prime n, there are only two 1-dimensional tilings, which give the known
solution for A363381 based on rows.  Still no indication if this is the only
solution.

Martin Fuller

PS The number of 1-dimensional tilings with reflections allowed is not yet
in OEIS.  The sequence without reflections is A067824.  I think the sequence
with reflections starts:
1, 2, 2, 4, 2, 8, 2, 13, 5, 20, 2, 54, 2, 68, 9, 160, 2, 287, 2, 582, 11,
1028, 2, 2315, 6, 4100, 17, 8454, 2, 16519, 2, 33469, 15, 65540, 11, 132915,
2, 262148, 17
I have half of a proof that there are 2^(k-1) tiles of size k which can tile
an interval of 2k.  It would be nice to get a proof and to get similar
relationships for T(3k,k), T(4k,k) etc.