OFFSET
1,1
COMMENTS
It is assumed that each of the n darts scores. - Colin Barker, May 19 2014
Starting at a(2) = 103, each subsequent term is 60 plus the previous term. Proof: All numbers from 2 to 102 can be scored using 2 darts. For 3 darts, the possible totals are the set of sums of the numbers between 2 and 102 (the 2-dart combinations) and the values for the third dart (see A242718 for allowable scores). Since the 2-dart scores are continuous in the 2 - 102 range, any value less than 102 plus the maximum 1-dart total can be obtained by selecting the maximum 1-dart score (60) and then choosing the (desired score - 60) from the 2-dart combinations. For example, to score 95 with 3 darts, assume dart 1 scores 60 and then choose 35 from the 2-dart score list. Thus, the lowest score that cannot be obtained with 3 darts is 61 (the maximum 1-dart score + 1) + 102 (the maximum 2-dart score) = 163. Repeat this approach for subsequent terms. - David Consiglio, Jr., Jun 11 2014
LINKS
Wikipedia, Darts
Index entries for linear recurrences with constant coefficients, signature (2,-1).
FORMULA
a(1) = 23, remaining terms: a(n) = 103 + 60*(n-2). - David Consiglio, Jr., Jun 11 2014
From Jianing Song, Jan 22 2021: (Start)
G.f.: 17 - 20*x + (77*x-17)/(1-x)^2.
E.g.f.: 17 - 20*x + (60*x-17)*exp(x). (End)
EXAMPLE
a(6) = 403. All numbers from 5 - 342 can be scored using 5 darts. Thus, 60 (dart 1) + 342 (remaining 5 darts) = 402 -> The maximum score for 6 darts. - David Consiglio, Jr., Jun 11 2014
MATHEMATICA
Join[{23}, NestList[60+#&, 103, 60]] (* Harvey P. Dale, Sep 18 2020 *)
PROG
(PARI) a(n) = 103 + 60*(n-2) - 20*!(n-1); \\ Jinyuan Wang, May 30 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
John Bibby, May 18 2014
EXTENSIONS
a(2)-a(4) from Colin Barker, May 19 2014
a(5)-a(11) from David Consiglio, Jr., Jun 12 2014
More terms from Harvey P. Dale, Sep 18 2020
STATUS
approved