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A301380
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Number of tied close American football games: number of ways for the game to have n scoring plays, never be separated by more than one score after each play, and be tied at the end.
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3
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1, 0, 14, 90, 1114, 10718, 113216, 1152540, 11906042, 122269186, 1258639394, 12943924960, 133168371652, 1369830663678, 14091618522696, 144958402357534, 1491181759508514, 15339664777115086, 157798158205312580, 1623258461571800764, 16698349602838663718, 171774768145224952472
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OFFSET
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0,3
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COMMENTS
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Each play (counting untimed downs as part of the previous play) can score at most 8 points for one team.
The same as counting walks that return to the x-axis of x-length n from the origin bounded above by y=8, below by y=-8, and using the steps {[1,8],..,[1,2],[1,-2],..,[1,-8]}.
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LINKS
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FORMULA
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G.f.: (1-4*t-45*t^2-43*t^3+98*t^4+108*t^5-24*t^6-30*t^7)/(1-4*t-59*t^2-77*t^3+170*t^4+234*t^5-92*t^6-142*t^7-4*t^8+6*t^9).
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EXAMPLE
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There are no tied games with 1 scoring play. To have tied games after 2 scoring plays requires each team to score the same number of points (7 possibilities) in each play (2 orderings): hence 14 walks.
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MAPLE
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taylor((1-4*t-45*t^2-43*t^3+98*t^4+108*t^5-24*t^6-30*t^7)/(1-4*t-59*t^2-77*t^3+170*t^4+234*t^5-92*t^6-142*t^7-4*t^8+6*t^9), t=0, N);
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CROSSREFS
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KEYWORD
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nonn,walk
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AUTHOR
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STATUS
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approved
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