OFFSET
0,6
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]
LINKS
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,0,1,1,1).
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
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Bob Selcoe, Feb 16 2014
STATUS
approved