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A133258 Number of possible 3 X n arrangements of black and white squares that can form the middle three rows in an n X n crossword puzzle with rotational symmetry. In this sequence, n is ODD. 1
1, 11, 38, 157, 718, 3039, 12571 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,2

COMMENTS

For this sequence, n must be odd.

LINKS

Table of n, a(n) for n=3..9.

MATHEMATICA

(*This program counts, lists and displays the possible three - row centers \ of an n X n (n odd) crossword puzzle with rotational symmetry.*)

plotnice = ArrayPlot [ #, Frame -> False, Mesh -> True, MeshStyle -> \

GrayLevel [ 0 ] ] &;

For [ w = 1, w <= 7, w++,

n = 2w + 1;

t = n - 3;

arrangements = {};

For [ r = 0, r <= t, r++,

m = Compositions [ n - r, r + 1 ];

m2 = Select [ m, FreeQ [ #, 2 ] & ];

m1 = Select [ m2, FreeQ [ #, 1 ] & ];

arrangements = Join [ arrangements, m1 ] ];

possiblecolumns = {};

For [ j = 1, j <= Length [ arrangements ], j++,

original = arrangements [ [ j ] ];

new = {};

For [ i = 1, i <= Length [ original ], i++, new = Append [ new,

Join [ Table [ 0, {original [ [ i ] ]} ], {1} ] ] ];

new = Drop [ Flatten [ new ], -1 ];

possiblecolumns = Append [ possiblecolumns, new ] ];

symmetricrows =

Select [ possiblecolumns,

possiblecolumns [ [ # ] ] == Reverse [ possiblecolumns [ [ # ] ] ] & ];

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 ];

reversedlegalrows = Reverse /@ legalrows;

potentialcenters = Flatten [ Table [ {legalrows [ [ i ] ], symmetricrows [ [ j ] ],

reversedlegalrows [ [ i ] ]}, {i, 1,

Length [ legalrows ]}, {j, 1, Length [ symmetricrows ]} ], 1 ];

transposedpotentialcenters = Transpose /@ potentialcenters;

freeof101s = Function [ FreeQ [ #, {1, 0, 1} ] ];

transposedno101s = Select [ transposedpotentialcenters, freeof101s ];

almostcenters = Transpose /@ transposedno101s;

insertzerorows =

Function [ Append [ Prepend [ #, Table [ 0, {n} ] ], Table [ 0, {n} ] ] ];

almostcenterswithzeros = insertzerorows /@ almostcenters;

centers = {};

centercount = 0;

For [ v = 1, v <= Length [ almostcenterswithzeros ], v++,

puzzlegraph = Table [ almostcenterswithzeros [ [ v, r, s ] ], {r, 1, 5}, {s,

1, n} ];

verts = {};

For [ i2 = 1, i2 <= 5, i2++, For [ j2 = 1, j2 <= n, j2++, If [ puzzlegraph [ [ i2,

j2 ] ] == 1, verts = Append [ verts, j2 + 5n - n*i2 ] ] ] ];

thegraph = DeleteVertices [ GridGraph [ n, 5 ], verts ];

If [ ConnectedQ [ thegraph ] ==

True, centercount = centercount + 1;

centers = Append [ centers, almostcenterswithzeros [ [ v ] ] ] ];

]

plotnice /@ centers;

Print [ "the number of center three-row arrangements in a ", n, " x ", n, " puzzle with rotational symmetry is ", centercount ];

Print [ " " ];

]

CROSSREFS

Cf. A130578.

Sequence in context: A007585 A024202 A213775 * A288745 A103738 A045801

Adjacent sequences:  A133255 A133256 A133257 * A133259 A133260 A133261

KEYWORD

nonn,more

AUTHOR

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

STATUS

approved

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Last modified March 19 23:02 EDT 2019. Contains 321343 sequences. (Running on oeis4.)