OFFSET
1,4
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.
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..5000
Fumio Hazama, Spectra of graphs attached to the space of melodies, Discr. Math., 311 (2011), 2368-2383. See Table 2.1.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,1,-1).
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
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
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.
MATHEMATICA
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"}}]
PROG
(Haskell)
a130578 n = a130578_list !! (n-1)
a130578_list = 0 : 0 : 1 : 3 : zipWith (+)
(map (* 2) $ drop 3 a130578_list)
(zipWith (-) (map (+ 1) a130578_list) (drop 2 a130578_list))
-- Reinhard Zumkeller, May 23 2013
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Marc A. Brodie (mbrodie(AT)wju.edu), Aug 10 2007, Aug 24 2007
STATUS
approved