3,2

In a standard American crossword puzzle, such as those in the New York Times, in any row there must be at least one run of white squares and all runs of white squares must be of length at least three.

a[n]=2a[n-1]-a[n-2]+a[n-3]+a[n-4]+f[n] where f[n]=b[n]^2-2b[n-1]^2+b[n-2]^2-b[n-3]^2-b[n-4]^2-2b[n-3] and b[n] is the sequence A130578

a[4]=9 = 3^2 because using 0's for white squares and 1's for black squares, the three possible rows in a 4 X 4 crossword are 0000, 1000 and 0001 and any of these three rows as a top row is compatible with any as a second row.

Furthermore, a[6]=98 < 100 = 10^2 because while 000111 and 111000 are two of the ten possible rows in a 6 X 6 crossword puzzle, the arrangement

000111

111000

would not be possible.

<< DiscreteMath`Combinatorica` (*This program counts, lists and displays the possible 2 - row patterns in an n X n crossword puzzle*)

plotnice = ArrayPlot [ #, Frame -> False, Mesh -> True, MeshStyle -> GrayLevel [ 0 ] ] &;

For [ n = 3, n <= 7, n++,

usablemods = {0, 1, 3, 7};

usablenumbers = Function [ MemberQ [ usablemods, Mod [ #, 8 ] ] ];

goodnumbers = Union [ Table [

k, {k, 0, 2^(n - 3) - 1} ], Table [ k, {k, 2^(n - 1), 2^n - 2} ] ];

numbers = Select [ goodnumbers, usablenumbers ];

rows = Table [ PadLeft [ IntegerDigits [ numbers [ [ j ] ], 2 ], n ], {j, 1, Length [

numbers ]} ];

no101s = Function [ FreeQ [ Partition [ #1, 3, 1 ], {1, 0, 1} ] ];

no1001s = Function [ FreeQ [ Partition [ #1, 4, 1 ], {1, 0, 0, 1} ] ];

legalrows = Select [ Select [ rows, no1001s ], no101s ];

tworows = Tuples [ legalrows, 2 ];

addrows = Function [ Plus [ # [ [ 1 ] ], # [ [ 2 ] ] ] ];

goodrows = Function [ Not [ FreeQ [ Plus [ # [ [ 1 ] ], # [ [ 2 ] ] ], 0 ] ] ];

goodtworows = Select [ tworows, goodrows ];

Print [ "the number of two-row arrangements in a ", n, " x ", n, " puzzle is \

", Length [ goodtworows ] ];

plotnice /@ goodtworows;

]

Cf. A130578.

Sequence in context: A128642 A022604 A085630 this_sequence A027602 A134537 A066647

Adjacent sequences: A133223 A133224 A133225 this_sequence A133227 A133228 A133229

nonn

Marc A. Brodie (mbrodie(AT)wju.edu), Jan 03 2008