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Search: id:A159257
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| A159257 |
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Rank deficiency of the size n "turn it red" problem. |
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+0
7
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| 0, 0, 0, 4, 2, 0, 0, 0, 8, 0, 6, 0, 0, 4, 0, 8, 2, 0, 16, 0, 0, 0, 14, 4, 0, 0, 0, 0, 10, 20, 0, 20, 16, 4, 6, 0, 0, 0, 32, 0, 2, 0, 0, 4, 0, 0, 30, 0, 8, 8, 0, 0, 2, 4, 0, 0, 0, 0, 22, 0, 40, 24, 0, 28, 42, 0, 32, 0, 8, 0, 14, 0, 0, 4, 0, 0, 2, 0, 64, 0, 0, 0, 6, 12, 0, 0, 0, 0, 10, 0, 0, 20, 0, 4, 62
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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A square array of n by n pixels can have two states (gray, red). Touching a pixel switches its state and the state of the adjacent pixels. The general problem consists in turning all pixels ON given any initial configuration. It requires inverting a nA^2 by nA^2 matrix in Z/2Z. The sequence is the rank deficiency of the matrix, such that the terms at 0 correspond to the sizes for which the general case admits a solution, and the others correspond to the dimension of the space of initial configurations that do not admit a solution.
The size 5 game can be played at the link given below. Rank deficiency is 2 for that game, but only initial configurations that admit a solution are given.
a(n) is non-zero iff n is in A117870; a(n) is zero iff n in A076436. [From Max Alekseyev (maxale(AT)gmail.com), Sep 17 2009]
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REFERENCES
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See A075462 for references. [From Max Alekseyev (maxale(AT)gmail.com), Sep 17 2009]
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LINKS
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Turn it red
Eric Weisstein's World of Mathematics, Lights Out Puzzle [From Max Alekseyev (maxale(AT)gmail.com), Sep 17 2009]
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EXAMPLE
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For n=2, matrix is [1 1 1 0][1 1 0 1][1 0 1 1][0 1 1 1] which is of full rank
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CROSSREFS
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Cf. A075462, A075463, A075464, A076436, A076437 [From Max Alekseyev (maxale(AT)gmail.com), Sep 17 2009]
Sequence in context: A105087 A028572 A107492 this_sequence A107088 A137986 A093486
Adjacent sequences: A159254 A159255 A159256 this_sequence A159258 A159259 A159260
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KEYWORD
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nonn
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AUTHOR
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Bruno Vallet (bruno.vallet(AT)gmail.com), Apr 07 2009
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EXTENSIONS
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More terms from Max Alekseyev (maxale(AT)gmail.com), Sep 17 2009
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