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A237997 Number of ordered ways to achieve a score of n in American football taking into account different scoring methods. 1
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..44.

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

Sequence in context: A239329 A094093 A240077 * A317885 A321535 A108809

Adjacent sequences:  A237994 A237995 A237996 * A237998 A237999 A238000

KEYWORD

nonn,easy

AUTHOR

Bob Selcoe, Feb 16 2014

STATUS

approved

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Last modified March 19 17:21 EDT 2019. Contains 321330 sequences. (Running on oeis4.)