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 A204092 The number of 1 by n Haunted Mirror Maze puzzles with a unique solution. 3
 1, 3, 17, 91, 449, 2123, 9841, 45211, 206881, 945003, 4313297, 19680571, 89784449, 409577483, 1868351281, 8522666971, 38876763361, 177338745003, 808940722577, 3690027171451, 16832256509249, 76781232397643, 350241657358321, 1597645838773531, 7287745912705441 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Requiring the last slot contain a mirror results in A204091.  Fixing orientation of mirrors results in A204090.  Fixing orientation and requiring the last slot be a mirror results in A204089. LINKS Samples of these types of puzzles can be found at this site. Index entries for linear recurrences with constant coefficients, signature (8,-19,16,-4). FORMULA a(n) = A204091(n+1)/2 - 2^(n+1) + 2. a(n) = 8*a(n-1) - 19*a(n-2) + 16*a(n-3) - 4*a(n-4), a(0) = 1, a(1) = 3, a(2) = 17, a(3) = 91. G.f.: (1-5x+12x^2-4x^3)/((1-x)(1-2x)(1-5x+2x^2)). EXAMPLE The following 17 boards illustrate K(2): ('Z', '/') ('Z', '|') ('V', '/') ('V', '|') ('G', 'G') ('G', '/') ('G', '|') ('/', 'Z') ('/', 'V') ('/', 'G') ('/', '/') ('/', '|') ('|', 'Z') ('|', 'V') ('|', 'G') ('|', '/') ('|', '|') MATHEMATICA LinearRecurrence[{8, -19, 16, -4}, {1, 3, 17, 91}, 40] PROG (Python) def a(n, d={0:1, 1:3, 2:17, 3:91}): .if n in d: ..return d[n] .d[n]=8*a(n-1) - 19*a(n-2) + 16*a(n-3) - 4*a(n-4) .return d[n] (Python) #Produces a(n) through enumeration and also displays boards: def Kprint(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 and '|' 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 or '|' in y: .....if '/' in y and '|' in y: ......m = min(y.index('/'), y.index('|')) .....else: ......if '/' in y: .......m=y.index('/') ......else: .......m=y.index('|') .....if y[0] not in ['/', '|']: #Don't need to add blank pieces to z ......z.append(y[:m]) .....y=y[m+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 CROSSREFS Cf. A204089, A204090, A204091. Sequence in context: A203851 A198766 A240652 * A221731 A290925 A020056 Adjacent sequences:  A204089 A204090 A204091 * A204093 A204094 A204095 KEYWORD nonn,easy AUTHOR David Nacin, Jan 10 2012 STATUS approved

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Last modified May 15 17:50 EDT 2021. Contains 343920 sequences. (Running on oeis4.)