

A241746


Smallest number greater than n that CANNOT be scored using n darts on a standard dartboard.


4



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

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 2dart combinations) and the values for the third dart (see A242718 for allowable scores). Since the 2dart scores are continuous in the 2  102 range, any value less than 102 plus the maximum 1dart total can be obtained by selecting the maximum 1dart score (60) and then choosing the (desired score  60) from the 2dart combinations. For example, to score 95 with 3 darts, assume dart 1 scores 60 and then choose 35 from the 2dart score list. Thus, the lowest score that cannot be obtained with 3 darts is 61 (the maximum 1dart score + 1) + 102 (the maximum 2dart score) = 163. Repeat this approach for subsequent terms.  David Consiglio, Jr., Jun 11 2014


LINKS

Table of n, a(n) for n=1..46.
Wikipedia, Darts
Index entries for linear recurrences with constant coefficients, signature (2,1).


FORMULA

a(1) = 23, remaining terms: a(n) = 103 + 60*(n2).  David Consiglio, Jr., Jun 11 2014
From Jianing Song, Jan 22 2021: (Start)
G.f.: 17  20*x + (77*x17)/(1x)^2.
E.g.f.: 17  20*x + (60*x17)*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*(n2)  20*!(n1); \\ Jinyuan Wang, May 30 2021


CROSSREFS

Sequence in context: A225584 A214092 A139976 * A142192 A129918 A240839
Adjacent sequences: A241743 A241744 A241745 * A241747 A241748 A241749


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



