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A187734
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a(n) is the number of n-walks between the vertices 1 and 3 of the Graph on the chalkboard in 'Good Will Hunting', (1997).
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0
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0, 2, 2, 14, 18, 94, 146, 638, 1138, 4382, 8658, 30398, 64818, 212574, 479890, 1496062, 3525106, 10581918, 25748306, 75139390, 187301554, 535144670, 1358396434, 3820058238, 9829858162, 27316621854, 71015537874, 195595836350, 512422576178, 1401935442782
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OFFSET
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1,2
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COMMENTS
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For a presentation of the Graph, see the first link.
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REFERENCES
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Burkard Polster & Marty Ross, Math Goes to the Movies, The Johns Hopkins University Press, Baltimore, 2013, ยง1.7 Mathematics: Graph Theory 1, pp. 9-12.
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LINKS
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Table of n, a(n) for n=1..30.
Oliver Knill, Harvard Math, The Good Will Hunting Problem
Wikipedia, Good Will Hunting
MMDB-The Mathematical Movie Database, Burkard Polster & Marty Ross, Good Will Hunting
Index entries for linear recurrences with constant coefficients, signature (1,6,-4).
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FORMULA
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G.f.: (2*x^2 + 2*x^3)/(1 - 7*x^2 - 2*x^3 + 4*x^4).
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EXAMPLE
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"For example, between the vertices 1 and 3, we can calculate that there are no 1-walks, two 2-walks, two 3-walks and so on. The resulting sequence of numbers begins 0, 2, 2, 14, 18, 94, 146, 638, ..." (p. 11).
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MATHEMATICA
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LinearRecurrence[{1, 6, -4}, {0, 2, 2}, 30] (* Or *)
Rest@ CoefficientList[ Series[ 2x^2/(1 - x - 6x^2 + 4x^3), {x, 0, 28}], x]
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PROG
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(PARI) Vec((2*x^2 + 2*x^3)/(1 - 7*x^2 - 2*x^3 + 4*x^4)+O(x^99)) \\ Charles R Greathouse IV, May 21 2013
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CROSSREFS
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Sequence in context: A032209 A032134 A032038 * A151353 A151437 A235349
Adjacent sequences: A187731 A187732 A187733 * A187735 A187736 A187737
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KEYWORD
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nonn,easy,walk
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AUTHOR
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Oliver Knill (knill(AT)math.harvard.edu), Burkard Polster (burkard.polster(AT)monash.edu), Marty Ross (martinirossi(AT)gmail.com), and Robert G. Wilson v, Jan 02 2013
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STATUS
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approved
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