A241746 Smallest number greater than n that CANNOT be scored using n darts on a standard dartboard.

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

Offset: 1

Author

John Bibby, May 18 2014

Keywords

• nonn
• easy

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

Examples

			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
		

Links

• Wikipedia, Darts
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Programs

Mathematica

Join[{23},NestList[60+#&,103,60]] (* Harvey P. Dale, Sep 18 2020 *)

PARI

a(n) = 103 + 60*(n-2) - 20*!(n-1); \\ Jinyuan Wang, May 30 2021

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)

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