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%I #24 Jun 13 2015 00:52:22
%S 0,0,1,3,6,10,16,26,43,71,116,188,304,492,797,1291,2090,3382,5472,
%T 8854,14327,23183,37512,60696,98208,158904,257113,416019,673134,
%U 1089154,1762288,2851442,4613731,7465175,12078908,19544084
%N Number of different possible rows (or columns) in an n X n crossword puzzle.
%C 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.
%C 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.
%H Reinhard Zumkeller, <a href="/A130578/b130578.txt">Table of n, a(n) for n = 1..5000</a>
%H Fumio Hazama, <a href="http://dx.doi.org/10.1016/j.disc.2011.06.008">Spectra of graphs attached to the space of melodies</a>, Discr. Math., 311 (2011), 2368-2383. See Table 2.1.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,1,-1).
%F 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.
%F 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])
%F O.g.f.: x^3/((-1+x)*(x^2+x-1)*(x^2-x+1)) . - _R. J. Mathar_, Nov 23 2007
%F a(n) = A005252(n+1) - 1. - _R. J. Mathar_, Nov 15 2011
%F G.f.: Q(0)*x^2/(2-2*x), where Q(k) = 1 + 1/(1 - x*( 4*k+2 -x +x^3)/( x*( 4*k+4 -x +x^3) +1/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jan 07 2014
%e 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.
%t 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"}}]
%o (Haskell)
%o a130578 n = a130578_list !! (n-1)
%o a130578_list = 0 : 0 : 1 : 3 : zipWith (+)
%o (map (* 2) $ drop 3 a130578_list)
%o (zipWith (-) (map (+ 1) a130578_list) (drop 2 a130578_list))
%o -- _Reinhard Zumkeller_, May 23 2013
%o (PARI) a(n)=([0,1,0,0,0;0,0,1,0,0;0,0,0,1,0;0,0,0,0,1;-1,1,1,-3,3]^n*[0;0;0;1;3])[1,1] \\ _Charles R Greathouse IV_, Jun 11 2015
%K nonn,easy
%O 1,4
%A Marc A. Brodie (mbrodie(AT)wju.edu), Aug 10 2007, Aug 24 2007
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