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%I #9 Oct 01 2023 13:14:10
%S 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,
%T 49,57,59,71,71,84,87,99,103,119,121,139,144,161,167,189,193,217,225,
%U 248,257,286,293,324,336,366,379,416,427,467,483,522,540,586,602,652,674
%N Expansion of 1/((1-x^2)(1-x^3)(1-x^6)(1-x^7)(1-x^8)).
%C 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).
%C 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).
%C Also number of (unordered) ways of making change for n cents with coins of sizes 2, 3, 6, 7 and 8 cents.
%H <a href="/index/Rec#order_26">Index entries for linear recurrences with constant coefficients</a>, 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).
%F G.f.: 1/((1-x^2)*(1-x^3)*(1-x^6)*(1-x^7)*(1-x^8))
%e 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).
%o (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),","))
%Y Cf. A069183.
%K nonn,easy
%O 0,7
%A _Rick L. Shepherd_, Apr 21 2002
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