

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
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OFFSET

0,6


COMMENTS

Alternate related equations include:
I.a) when n != 0 (mod 4): a(n) = a(n2) + a(n3) + a(n4)
I.b) when n == 0 (mod 8): a(n) = a(n2) + a(n3) + a(n4) + 1
I.c) when n == 4 (mod 8): a(n) = a(n2) + a(n3) + a(n4)  1
II.a) when n == 4..7 (mod 8): a(n) = a(n1) + a(n3)
II.b) when n == {0,2}(mod 8): a(n) = a(n1) + a(n3) + 1
II.c) when n == {1,3} (mod 8): a(n) = a(n1) + a(n3)  1
The sequence applies only when considering HOW points are scored. When not taking this into account (i.e., safety and twopoint 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)*(1x^3x)*(x^4+1) ).
a(n) = a(n2) + a(n3) + a(n6) + a(n7) + a(n8).
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(n4).  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 twopoint 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



