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A338551 Number of ways to make a checkout score of n in darts.

%I A338551 #33 Nov 12 2020 01:17:16
%S A338551 0,1,1,4,7,14,20,31,39,55,65,86,96,126,133,171,179,223,228,286,283,
%T A338551 352,348,422,408,497,467,569,534,642,594,720,654,791,719,863,775,942,
%U A338551 831,1012,894,1082,945,1159,991,1216,1037,1263,1062,1311,1081,1340,1110,1366
%N A338551 Number of ways to make a checkout score of n in darts.
%C A338551 In other words, the number of ways to achieve a score of n using at most 3 darts and finishing on a double. The maximum checkout score is 170, so this is a finite sequence.
%H A338551 Carmen Bruni, <a href="/A338551/b338551.txt">Table of n, a(n) for n = 1..170</a>
%H A338551 Wikipedia, <a href="https://en.wikipedia.org/wiki/Darts">Darts</a>
%o A338551 (Python)
%o A338551 def darts(n):
%o A338551   if n > 170 or n <= 1:
%o A338551     return 0
%o A338551   ans = 0
%o A338551   singles = list(range(1, 21)) + [25]
%o A338551   doubles = list(map(lambda x: 2*x, singles))
%o A338551   triples = list(map(lambda x: 3*x, singles[:-1]))
%o A338551   throws = singles+doubles+triples
%o A338551   for i in range(len(throws)):
%o A338551     for j in range(len(throws)):
%o A338551       for k in range(len(doubles)):
%o A338551         dart1 = throws[i]
%o A338551         dart2 = throws[j]
%o A338551         dart3 = doubles[k]
%o A338551         if dart1 + dart2 + dart3 == n:
%o A338551           ans += 1
%o A338551     for j in range(len(doubles)):
%o A338551       dart1 = throws[i]
%o A338551       dart2 = doubles[j]
%o A338551       if dart1 + dart2 == n:
%o A338551         ans += 1
%o A338551   return ans + (n in doubles)
%o A338551 for i in range(1,171):
%o A338551   print(darts(i))
%o A338551 (PARI) seq()={my(s=x*(1-x^20)/(1-x)+x^25, d=subst(s,x,x^2), g=s+d+subst(s-x^25,x,x^3)); Vecrev((1+g+g^2)*d/x)} \\ _Andrew Howroyd_, Nov 04 2020
%Y A338551 Cf. A008575, A242718, A242678, A242681, A167213, A076119, A244512, A117883.
%K A338551 nonn,fini,full
%O A338551 1,4
%A A338551 _Carmen Bruni_, Nov 02 2020