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A241746
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Smallest number greater than n that CANNOT be scored using n darts on a standard dartboard.
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4
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23, 103, 163, 223, 283, 343, 403, 463, 523, 583, 643, 703, 763, 823, 883, 943, 1003, 1063, 1123, 1183, 1243, 1303, 1363, 1423, 1483, 1543, 1603, 1663, 1723, 1783, 1843, 1903, 1963, 2023, 2083, 2143, 2203, 2263, 2323, 2383, 2443, 2503, 2563, 2623, 2683, 2743
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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G.f.: 17 - 20*x + (77*x-17)/(1-x)^2.
E.g.f.: 17 - 20*x + (60*x-17)*exp(x). (End)
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EXAMPLE
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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
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MATHEMATICA
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Join[{23}, NestList[60+#&, 103, 60]] (* Harvey P. Dale, Sep 18 2020 *)
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PROG
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(PARI) a(n) = 103 + 60*(n-2) - 20*!(n-1); \\ Jinyuan Wang, May 30 2021
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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