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 A204090 The number of 1 X n Haunted Mirror Maze puzzles with a unique solution where mirror orientation is fixed. 4
 1, 2, 8, 34, 134, 498, 1786, 6274, 21778, 75074, 257762, 882946, 3020354, 10323714, 35270530, 120467458, 411394306, 1404773378, 4796567042, 16377245698, 55916897282, 190915194882, 651831179266, 2225502715906, 7598365282306, 25942489251842, 88573293551618 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Since the uniqueness of a solution is unaffected by the orientation of the mirrors in this 1 X n case, we assume mirror orientation is fixed for this sequence. Connected to A204089, which counts the 1 X n boards with unique solutions that end in a mirror. Dropping the mirror orientation restriction would give A204092. Dropping the orientation restriction and requiring a mirror in the last slot gives A204091. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Samples of these types of puzzles can be found at this and other sites. Index entries for linear recurrences with constant coefficients, signature (7,-16,14,-4). FORMULA G.f.: (1 - 5*x + 10*x^2 - 4*x^3) / ((1 - x)*(1 - 2*x)*(1 - 4*x + 2*x^2)). a(n) = A204089(n+1) - 2^(n+1) + 2. a(n) = 7*a(n-1) - 16*a(n-2) + 14*a(n-3) - 4*a(n-4), a(0)=1, a(1)=2, a(2)=8, a(3)=34. a(n) = 2 - 2^(1+n) + ((2+sqrt(2))^(1+n) - (2-sqrt(2))^(1+n))/(2*sqrt(2)). - Colin Barker, Nov 26 2016 EXAMPLE For n=3 we would get the following 34 boards with unique solutions: ('Z', 'Z', '/') ('Z', 'G', '/') ('Z', '/', 'Z') ('Z', '/', 'V') ('Z', '/', 'G') ('Z', '/', '/') ('V', 'V', '/') ('V', 'G', '/') ('V', '/', 'Z') ('V', '/', 'V') ('V', '/', 'G') ('V', '/', '/') ('G', 'Z', '/') ('G', 'V', '/') ('G', 'G', 'G') ('G', 'G', '/') ('G', '/', 'Z') ('G', '/', 'V') ('G', '/', 'G') ('G', '/', '/') ('/', 'Z', 'Z') ('/', 'Z', 'G') ('/', 'Z', '/') ('/', 'V', 'V') ('/', 'V', 'G') ('/', 'V', '/') ('/', 'G', 'Z') ('/', 'G', 'V') ('/', 'G', 'G') ('/', 'G', '/') ('/', '/', 'Z') ('/', '/', 'V') ('/', '/', 'G') ('/', '/', '/') MATHEMATICA LinearRecurrence[{7, -16, 14, -4}, {1, 2, 8, 34}, 40] PROG (Python) def a(n, d={0:1, 1:2, 2:8, 3:34}): .if n in d: ..return d[n] .d[n]=7*a(n-1) - 16*a(n-2) + 14*a(n-3) - 4*a(n-4) .return d[n] (Python) #Produces a(n) through enumeration and also displays boards: def Hprint(n): .print('The following generate boards with a unique solution') .s=0 .for x in product(['Z', 'V', 'G', '/'], repeat=n): ..#Taking care of the no mirror case ..if '/' not in x: ...if 'Z' not in x and 'V' not in x: ....s+=1 ....print(x) ..else: ...#Splitting x up into a list pieces ...y=list(x) ...z=list() ...while y: ....if '/' in y: .....if y[0] != '/': #Don't need to add blank pieces to z ......z.append(y[:y.index('/')]) .....y=y[y.index('/')+1:] ....else: .....z.append(y) .....y=[] ...#For each element in the list checking for Z&V together ...goodword=True ...for w in z: ....if 'Z' in w and 'V' in w: .....goodword=False ...if goodword: ....s+=1 ....print(x) .return s (PARI) Vec((1-5*x+10*x^2-4*x^3) / ((1-x)*(1-2*x)*(1-4*x+2*x^2)) + O(x^30)) \\ Colin Barker, Nov 26 2016 CROSSREFS Cf. A204089, A204091, A204092. Sequence in context: A172448 A268601 A026577 * A226495 A111643 A000163 Adjacent sequences:  A204087 A204088 A204089 * A204091 A204092 A204093 KEYWORD nonn,easy AUTHOR David Nacin, Jan 10 2012 STATUS approved

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Last modified December 8 14:38 EST 2019. Contains 329865 sequences. (Running on oeis4.)