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A003213
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Number of ways to quarter a 2n X 2n chessboard.
(Formerly M3987)
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4
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OFFSET
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0,3
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COMMENTS
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Warning: it now seems very likely that this is an incorrect version of A257952. - N. J. A. Sloane, Apr 17 2016
Number of ways to dissect a 2n X 2n chessboard into 4 congruent pieces.
One can ask the same question for a 2n+1 X 2n+1 board if one omits the center square: this gives A006067.
a(0)=1, since there is one way to do nothing.
This sequence is wrong because of a bug in Mr. Parkin's code, and amazingly I can pinpoint exactly what the bug is! (I can reproduce his results.)
Firstly the description of the problem and its solution in Mr. Parkin's letter is very clear -- he doesn't leave a lot of room for misinterpretation (this is hugely to his credit). He also includes a very clear description of his algorithm, so I decided I would just code it up. I obtained Giovanni Resta's results as given in A257952 -- there is nothing wrong with Mr Parkin's algorithm.
A detailed breakdown of Parkin's results is also provided in the letter. All the results match with the exception of the final line. (This would be highly improbable if there was a completely different interpretation.) In any case, one sentence stood out as a possible red flag: "Further, there are potential mirror image paths in both cases when starting on the centre lines and these are prevented by requiring a turn in one direction on the path prior to allowing a turn in the other direction" (bottom of page 6). The discrepancy in results does indeed relate to the center line and if I modify my code to lose the flag on recursion, then I get Mr. Parkin's results (so turn in one direction is only prohibited for one step). (End)
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REFERENCES
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M. Gardner, The Unexpected Hanging and Other Mathematical Diversions. Simon and Schuster, NY, 1969, p. 189.
Popular Computing (Calabasas, CA), Vol. 1 (No. 7, 1973), Problem 15, front cover and page 2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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