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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.)