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A130578
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Number of different possible rows (or columns) in an n X n crossword puzzle.
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11
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0, 0, 1, 3, 6, 10, 16, 26, 43, 71, 116, 188, 304, 492, 797, 1291, 2090, 3382, 5472, 8854, 14327, 23183, 37512, 60696, 98208, 158904, 257113, 416019, 673134, 1089154, 1762288, 2851442, 4613731, 7465175, 12078908, 19544084
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| The number of linear arrangements of n black and white squares subject to the conditions that there must be at least one run of white squares and all runs of white squares must be of length at least three.
Crossword puzzles such as those in the New York Times do not include one-letter or two-letter words. Since the daily NYT puzzle is 15 X 15, there are a(15) = 797 different possible arrangements for each row.
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REFERENCES
| Fumio Hazama, Spectra of graphs attached to the space of melodies, Discr. Math., 311 (2011), 2368-2383. See Table 2.1.
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (3,-3,1,1,-1).
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FORMULA
| Recurrence: a[n + 4] = 2 a[n + 3] - a[n + 2] + a[n] + 1, with a[1] = 0, a[2] = 0, a[3] = 1, a[4] = 3.
Formula: a[n] = (30 - 30*Sqrt[5] - 30*(1/2 - Sqrt[5]/2)^n + 12*Sqrt[5]*(1/2 - Sqrt[5]/2)^n + 15*(1/2 + Sqrt[5]/2)^n + 3*Sqrt[5]*(1/2 + Sqrt[5]/2)^n - 15*Cos[(n*Pi)/3] + 15*Sqrt[5]*Cos[(n*Pi)/3] + 5*Sqrt[3]*Sin[(n*Pi)/3] - 5*Sqrt[15]*Sin[(n*Pi)/3])/(30*(-1 + Sqrt[5])
O.g.f.: x^3/((-1+x)*(x^2+x-1)*(x^2-x+1)) . - R. J. Mathar, Nov 23 2007
a(n) = A005252(n+1)-1. - R.J. Mathar, Nov 15 2011
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EXAMPLE
| a(5) = 6 because using 0's for white squares and 1's for black, the possible rows are: 00011, 10001, 11000, 00001, 10000, 00000
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MATHEMATICA
| Clear[k]; Clear[n]; possiblerows = {}; For[n = 1, n <= 36, n++, table = Table[{n, k, Coefficient[(x^0 + Sum[x^i, {i, 3, n - k}])^(k + 1), x, n - k]}, {k, 0, n}]; total = Sum[table[[j, 3]], {j, 1, n}]; possiblerows = Append[possiblerows, total]; totalstable = Table[{t, possiblerows[[t]]}, {t, 1, Length[ possiblerows]}]]; TableForm[totalstable, TableHeadings -> {None, {" n = squares", "total number of permissible rows"}}]
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CROSSREFS
| Sequence in context: A152009 A114324 A054886 * A107068 A033541 A038505
Adjacent sequences: A130575 A130576 A130577 * A130579 A130580 A130581
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KEYWORD
| nonn
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AUTHOR
| Marc A. Brodie (mbrodie(AT)wju.edu), Aug 10 2007, Aug 24 2007
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