cp's OEIS Frontend

A006067 Number of ways to quarter an n X n chessboard, with the central square removed for odd n.

%I A006067 M3769 #47 Jun 17 2025 00:45:06
%S A006067 1,1,1,5,7,37,104,766,3970,43318,431932,7695805,137066448,4015896016,
%T A006067 128095791922,6371333036059,355704307903818,30153126159555641,
%U A006067 2952926822418475378,431453249608567040694,73569487283165427567144,18558756256964594960321428
%N A006067 Number of ways to quarter an n X n chessboard, with the central square removed for odd n.
%C A006067 To "quarter" means to dissect in 4 parts, identical up to rotation, whose interior must be connected. (I.e., the parts must be polyominoes, any 1 X 1 square of which must share a side with some other 1 X 1 square of the part, unless there's only one.) Solutions that differ only by rotation or reflection are not counted separately.
%C A006067 See A257952 for much more information.
%C A006067 See A272070 for information on odd terms.
%D A006067 M. Gardner, The Unexpected Hanging and Other Mathematical Diversions. Simon and Schuster, NY, 1969, p. 189.
%D A006067 T. R. Parkin, personal communication.
%D A006067 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006067 Andrew Howroyd, <a href="/A006067/b006067.txt">Table of n, a(n) for n = 1..28</a>
%H A006067 Audrey Gruenberger (Editor), <a href="https://archive.org/details/Popular_Computing/Popular%20Computing%207/mode/1up">Checkerboard - Problem 15</a>, Popular Computing, Vol. 1, No. 7 (1973), front cover and page 2. (<a href="/A003213/a003213.jpg">Local copy of a scan of the cover illustration showing the a(6)-1 = 36 nontrivial solutions for n = 6, omitting the trivial solution using four squares.</a>)
%F A006067 a(2n) = A257952(n), a(2n+1) = A272070(n). - _Andrew Howroyd_, Apr 19 2016
%e A006067 For n = 1, we have the 1 X 1 board of which we remove the central square, so nothing is left, and the empty tiling is the only possible tiling.
%e A006067 For n = 2, we have the 2 X 2 board which can only be quartered using four 1 X 1 squares, so a(2) = 1 as well.
%e A006067 For n = 3, the 3 X 3 board without the central square can only be quartered using four 2 X 1 rectangles, so a(3) = 1 as well. (The two different solutions where the top rectangle is aligned to the left or to the right are counted as one, since they only differ by reflection.)
%e A006067 For n = 4 there is the trivial solution using squares, one using straight 4 X 1 tiles, one using T-shaped tiles, and two non-isomorphic ones using L-shaped tiles, one with a central symmetry and one with an axial symmetry:
%e A006067           A A B B       A B C D       A B B B        A A B B        A A B B
%e A006067   square: A A B B   I:  A B C D   T:  A A B C   Lc:  A C B D   La:  A C D B
%e A006067           C C D D       A B C D       A D C C        A C B D        A C D B
%e A006067           C C D D       A B C D       D D D C        C C D D        C C D D
%Y A006067 Cf. A257952, A064941, A272070.
%K A006067 nonn,nice
%O A006067 1,4
%A A006067 _N. J. A. Sloane_
%E A006067 a(8) corrected, a(9)-a(22) from _Andrew Howroyd_, Apr 18 2016
%E A006067 Name edited to clarify definition for odd n, and other edits by _M. F. Hasler_, Jun 13 2025