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A237997
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Number of ordered ways to achieve a score of n in American football taking into account different scoring methods.
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1
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1, 0, 1, 1, 1, 2, 3, 4, 7, 9, 14, 20, 29, 43, 63, 92, 136, 198, 291, 426, 624, 915, 1341, 1965, 2881, 4221, 6187, 9067, 13288, 19475, 28542, 41830, 61306, 89847, 131678, 192983, 282830, 414508, 607491, 890321, 1304830, 1912320, 2802642, 4107471, 6019791
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OFFSET
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0,6
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COMMENTS
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Alternate related equations include:
I.a) when n != 0 (mod 4): a(n) = a(n-2) + a(n-3) + a(n-4)
I.b) when n == 0 (mod 8): a(n) = a(n-2) + a(n-3) + a(n-4) + 1
I.c) when n == 4 (mod 8): a(n) = a(n-2) + a(n-3) + a(n-4) - 1
II.a) when n == 4..7 (mod 8): a(n) = a(n-1) + a(n-3)
II.b) when n == {0,2}(mod 8): a(n) = a(n-1) + a(n-3) + 1
II.c) when n == {1,3} (mod 8): a(n) = a(n-1) + a(n-3) - 1
The sequence applies only when considering HOW points are scored. When not taking this into account (i.e., safety and two-point conversion are considered indistinguishable because both are worth two points), then the sequence is A160993.
Number of compositions of n into parts 2, 3, 6, 7, and 8. [Joerg Arndt, Feb 18 2014]
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LINKS
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FORMULA
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G.f.: 1 / ( (1+x)*(1-x^3-x)*(x^4+1) ).
a(n) = a(n-2) + a(n-3) + a(n-6) + a(n-7) + a(n-8).
6*a(n) = 2*A068921(n) + (-1)^n +b(n) where b(n) = 3,-1,1,1,-3,1..., n>=0 is periodic with b(n) = -b(n-4). - R. J. Mathar, Mar 20 2017
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EXAMPLE
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a(8) = 7 because there are seven ways to score a total of 8 points: (a) touchdown and two-point conversion, (b) two field goals and a safety (3 orders), (c) a touchdown and safety (2 orders), and (d) four safeties.
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MATHEMATICA
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CoefficientList[Series[1/((1 + x) (1 - x^3 - x) (x^4 + 1)), {x, 0, 44}], x] (* or *)
LinearRecurrence[{0, 1, 1, 0, 0, 1, 1, 1}, {1, 0, 1, 1, 1, 2, 3, 4, 7}, 45] (* Michael De Vlieger, Mar 20 2017 *)
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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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