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A069745
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Expansion of 1/((1-x^2)(1-x^3)(1-x^6)(1-x^7)(1-x^8)).
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1
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1, 0, 1, 1, 1, 1, 3, 2, 4, 4, 5, 5, 8, 7, 11, 11, 14, 14, 19, 18, 24, 25, 30, 31, 39, 38, 47, 49, 57, 59, 71, 71, 84, 87, 99, 103, 119, 121, 139, 144, 161, 167, 189, 193, 217, 225, 248, 257, 286, 293, 324, 336, 366, 379, 416, 427, 467, 483, 522, 540, 586, 602, 652, 674
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OFFSET
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0,7
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COMMENTS
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Number of ways for a team to score n points in American football, where, except as noted below, order of scoring does not matter. Points can be scored as follows: 2 points for a safety (S), 3 points for a field goal (FG), 6 points for a touchdown (TD), 7 points for a touchdown followed by the point-after-touchdown (TD+PAT), 8 points for a touchdown followed by the 2-point-conversion (TD+2PC).
Touchdowns, field goals and safeties are scored independently of each other. Note, though, that a lone point cannot be scored unless a touchdown has just been scored by the same team. Similarly, a 2-point-conversion can only follow a touchdown. If these two restrictions were not part of the rules, then the U.S.A. football score sequence would be given instead by A069183 (with independent 1 and two independent types of 2s).
Also number of (unordered) ways of making change for n cents with coins of sizes 2, 3, 6, 7 and 8 cents.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 0, -1, 1, 1, 0, -2, -2, 0, 1, 0, -1, 0, 2, 2, 0, -1, -1, 1, 0, -1, -1, 0, 1).
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FORMULA
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G.f.: 1/((1-x^2)*(1-x^3)*(1-x^6)*(1-x^7)*(1-x^8))
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EXAMPLE
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a(14)=11: Here are the 11 ways that 14 points can be scored (although some scenarios, especially involving safeties, are unlikely in actual games): TD+PAT+TD+PAT ((6+1)+(6+1)), TD+TD+S (6+6+2), TD+TD+2PC (6+(6+2)), TD+FG+FG+S (6+3+3+2), TD+2PC+FG+FG ((6+2)+3+3), TD+PAT+FG+S+S ((6+1)+3+2+2), TD+2PC+S+S+S ((6+2)+2+2+2), TD+S+S+S+S (6+2+2+2+2), FG+FG+FG+FG+S (3+3+3+3+2), FG+FG+S+S+S+S (3+3+2+2+2+2) and S+S+S+S+S+S+S (2+2+2+2+2+2+2).
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PROG
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(PARI) a(n)=polcoeff(1/((1-x^2)*(1-x^3)*(1-x^6)*(1-x^7)*(1-x^8)+x*O(x^n)), n) for(n=0, 100, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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