# [seqfan] Chessboard polyominoes

Joseph S. Myers jsm at polyomino.org.uk
Sun Nov 28 02:51:12 CET 2010

Continuing my computations of various sequences related to various forms
of generalised polyominoes, I tried to compute the sequences for
"chessboard polyominoes" - A001933 and A001071.

The main reference for both those sequences (whose terms are unchanged
from the 1973 book) is "W. F. Lunnon, personal communication" so I am just
going by the names of the sequences to interpret what they mean.  The
natural interpretation would be polyominoes with n squares cut from a
chessboard (i.e. an infinite chessboard pattern), where for A001933
rotating and turning over the polyomino is not considered to result in a
different shape (and the reverse of the chessboard is supposed to have the
same pattern) and for A001071 rotating is not considered to result in a
different shape but two polyominoes related only by a reflection are
considered different.

Unfortunately this doesn't reproduce the terms given for A001071.  For
A001933 things are fine; I get 2, 1, 4, 7, 24, 62, 216, 710, 2570, 9215,
34146, 126853, 477182, 1802673, 6853152, 26153758, 100215818 which agrees
with the given terms and adds four more.  But for my interpretation of
A001071 I get 2, 1, 4, 10, 36, 110, 392, 1371, 5000, 18251, 67792, 253040,
952540, 3602846, 13699554, 52298057, 200406388, which disagrees on the
even-numbered terms only from a(6) onwards.

I wonder in this case if the sequence is actually meant to be counting
something else.  In particular, perhaps two one-sided chessboard
polyominoes are also (in A001071) considered the same if the shapes of the
two polyominoes (ignoring the colours) are the same or related by a
rotation or translation, but the colourings are only related by a
reflection.  This yields the value of a(6) in the sequence, since the
relevant cases are (in ASCII art)

XO
O
X
OX

X
OX
XO
O

where a reflection swaps the colours and I think it also reproduces a(8).
If Neil still has Lunnon's letter it might help elucidate what was meant.
The rule I suggest above is outside the range of rules my polyform
enumeration code can handle at present, but it would be interesting if
anyone can implement something that reproduces the terms given in the
sequence.

As for a sequence for the terms I computed for what I consider the natural
interpretation of "one-sided chessboard polyominoes", the definition of
A121198 is rather confusingly explained - that sequence is marked "obsc",
but as best I understand it I think it is equivalent to what I computed as
one-sided chessboard polyominoes, and so I think that A-number (with a
rewritten description) will be the natural home for my terms.

--
Joseph S. Myers
jsm at polyomino.org.uk