cp's OEIS Frontend

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

%I A237997 #22 Mar 20 2017 09:17:18
%S A237997 1,0,1,1,1,2,3,4,7,9,14,20,29,43,63,92,136,198,291,426,624,915,1341,
%T A237997 1965,2881,4221,6187,9067,13288,19475,28542,41830,61306,89847,131678,
%U A237997 192983,282830,414508,607491,890321,1304830,1912320,2802642,4107471,6019791
%N A237997 Number of ordered ways to achieve a score of n in American football taking into account different scoring methods.
%C A237997 Alternate related equations include:
%C A237997 I.a) when n != 0 (mod 4): a(n) = a(n-2) + a(n-3) + a(n-4)
%C A237997 I.b) when n == 0 (mod 8): a(n) = a(n-2) + a(n-3) + a(n-4) + 1
%C A237997 I.c) when n == 4 (mod 8): a(n) = a(n-2) + a(n-3) + a(n-4) - 1
%C A237997 II.a) when n == 4..7 (mod 8):  a(n) = a(n-1) + a(n-3)
%C A237997 II.b) when n == {0,2}(mod 8): a(n) = a(n-1) + a(n-3) + 1
%C A237997 II.c) when n == {1,3} (mod 8): a(n) = a(n-1) + a(n-3) - 1
%C A237997 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.
%C A237997 Number of compositions of n into parts 2, 3, 6, 7, and 8. [_Joerg Arndt_, Feb 18 2014]
%H A237997 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,0,0,1,1,1).
%F A237997 G.f.: 1 / ( (1+x)*(1-x^3-x)*(x^4+1) ).
%F A237997 a(n) = a(n-2) + a(n-3) + a(n-6) + a(n-7) + a(n-8).
%F A237997 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
%e A237997 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.
%t A237997 CoefficientList[Series[1/((1 + x) (1 - x^3 - x) (x^4 + 1)), {x, 0, 44}], x] (* or *)
%t A237997 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 *)
%K A237997 nonn,easy
%O A237997 0,6
%A A237997 _Bob Selcoe_, Feb 16 2014