A237997 Number of ordered ways to achieve a score of n in American football taking into account different scoring methods.

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

Offset: 0

Author

Bob Selcoe, Feb 16 2014

Keywords

• nonn
• easy

Comments

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]

Examples

			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.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,0,1,1,1).

Programs

Mathematica

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 *)

Formula

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