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A334288 Number of tieless rugby (union) games with n scoring events. 2
1, 6, 30, 180, 1002, 6012, 34224, 205344, 1180010, 7080060, 40911324, 245467944, 1423944024, 8543664144, 49710351720, 298262110320, 1739627237002, 10437763422012, 61002039226716, 366012235360296, 2142786218045748, 12856717308274488, 75380119335678608 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
In rugby (union) a scoring event can give 3, 5 or 7 points.
In April 1992 the current scoring format was introduced: 3 points are awarded for kicks/penalties, 5 points for unconverted tries and 7 points for converted tries. A game is a list of members of {-7,-5,-3,3,5,7} with negative points for the away team, positive for the home team.
A tieless game is one in which the teams never have the same score (except at the beginning, when no team has scored yet).
LINKS
EXAMPLE
a(2)=30, because there are 6^2=36 sequences of length 2 from {3,5,7,-3,-5,-7}; the 6 sequences that correspond to games with ties are precisely those of the form {k,-k}.
PROG
(Python)
def number_of_tieless_rugby_games(n):
"""
Returns the number of tieless rugby games with n scoring events.
A scoring event is a number in (-7, -5, -3, 3, 5, 7) and a game is tieless
if the score is never zero, apart from at the start.
Negative points represent points for the away team, positive points
represent points for the home team
"""
dictionary_of_scores = {0:1}
# The keys of this dictionary represent possible scores.
# The values represent the number of ways this score can be reached.
scoring_events = (-7, -5, -3, 3, 5, 7)
for i in range(n):
# At each stage, we have the nonzero scores with i scoring events in
# dictionary_of_scores. To find nonzero scores with i+1 scoring events
# consider each nonzero score, and each possibility for the next
# scoring event.
old_dictionary = dictionary_of_scores
dictionary_of_scores = {}
for score, number_of_ways in old_dictionary.items():
for scoring_event in scoring_events:
new_score = score + scoring_event
if new_score != 0:
dictionary_of_scores[new_score] =\
dictionary_of_scores.get(new_score, 0) + number_of_ways
return sum(dictionary_of_scores.values())
CROSSREFS
Inspired by A137684.
Sequence in context: A089896 A057754 A001473 * A063888 A029571 A357599
KEYWORD
nonn
AUTHOR
Cameron Ford, Jun 13 2020
STATUS
approved

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Last modified April 14 18:28 EDT 2024. Contains 371667 sequences. (Running on oeis4.)