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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

Table of n, a(n) for n=0..24.

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.)