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A318271 The optimum crossing time for the Bridge and Torch problem, given that the crossing times for the group's members are given by the n-th partition in A026791.

%I A318271 #26 Sep 04 2023 10:59:45
%S A318271 1,1,2,3,2,3,5,4,3,2,4,7,6,5,5,4,3,5,9,8,7,6,6,6,5,6,4,3,6,11,10,9,8,
%T A318271 8,7,7,8,7,7,6,7,5,4,7,13,12,11,10,10,9,9,9,8,7,8,9,8,8,7,10,8,8,6,5,
%U A318271 4,8,15,14,13,12,12,11,11,11,10,9,10,10,9,8,9
%N A318271 The optimum crossing time for the Bridge and Torch problem, given that the crossing times for the group's members are given by the n-th partition in A026791.
%H A318271 User baseman101, <a href="https://codegolf.stackexchange.com/q/75615/53884">The Bridge and Torch Problem</a>, Programming Puzzles & Code Golf Stack Exchange.
%H A318271 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bridge_and_torch_problem">Bridge and torch problem</a>.
%e A318271 When the crossing times are [1,2,5,10], the minimum total time for the group to cross is 17 minutes:
%e A318271   (2m)  1 and 2 cross,
%e A318271   (1m)  1 returns,
%e A318271   (10m) 5 and 10 cross,
%e A318271   (2m)  2 returns,
%e A318271   (2m)  1 and 2 cross.
%e A318271 +----+--------------------+------+
%e A318271 |  n | Crossing times     | a(n) |
%e A318271 +----+--------------------+------+
%e A318271 |  1 | [1]                |  1   |
%e A318271 |  2 | [1, 1]             |  1   |
%e A318271 |  3 | [2]                |  2   |
%e A318271 |  4 | [1, 1, 1]          |  3   |
%e A318271 |  5 | [1, 2]             |  2   |
%e A318271 |  6 | [3]                |  3   |
%e A318271 |  7 | [1, 1, 1, 1]       |  5   |
%e A318271 |  8 | [1, 1, 2]          |  4   |
%e A318271 |  9 | [1, 3]             |  3   |
%e A318271 | 10 | [2, 2]             |  2   |
%e A318271 | 11 | [4]                |  4   |
%e A318271 | 12 | [1, 1, 1, 1, 1]    |  7   |
%e A318271 | 13 | [1, 1, 1, 2]       |  6   |
%e A318271 | 14 | [1, 1, 3]          |  5   |
%e A318271 | 15 | [1, 2, 2]          |  5   |
%e A318271 | 16 | [1, 4]             |  4   |
%e A318271 | 17 | [2, 3]             |  3   |
%e A318271 | 18 | [5]                |  5   |
%e A318271 | 19 | [1, 1, 1, 1, 1, 1] |  9   |
%e A318271 | 20 | [1, 1, 1, 1, 2]    |  8   |
%e A318271 | 21 | [1, 1, 1, 3]       |  7   |
%e A318271 | 22 | [1, 1, 2, 2]       |  6   |
%e A318271 | 23 | [1, 1, 4]          |  6   |
%e A318271 | 24 | [1, 2, 3]          |  6   |
%e A318271 | 25 | [1, 5]             |  5   |
%e A318271 | 26 | [2, 2, 2]          |  6   |
%e A318271 | 27 | [2, 4]             |  4   |
%e A318271 | 28 | [3, 3]             |  3   |
%e A318271 | 29 | [6]                |  6   |
%e A318271 +----+--------------------+------+
%o A318271 (Julia)
%o A318271 function BT(p)
%o A318271     n = length(p)
%o A318271     p[end] = -(sum(p) + (n > 2 ? (n-3) * p[1] : 0))
%o A318271     if n >= 3
%o A318271         q = 2p[2] - p[1]; tog = false
%o A318271         for k in n-1:-1:1
%o A318271             (tog = ~tog) && p[k] > q ? p[k] -= q : p[k] = 0
%o A318271         end
%o A318271     end
%o A318271 -sum(p) end
%o A318271 [BT(p) for n in 1:9 for p in A026791(n)] |> println # _Peter Luschny_, Oct 18 2019
%Y A318271 Cf. A026791, A078476.
%K A318271 nonn,nice
%O A318271 1,3
%A A318271 _Peter Kagey_, Aug 22 2018
%E A318271 Terms a(45) and beyond added using Erwan's program from CodeGolf StackExchange by _Andrey Zabolotskiy_, Oct 18 2019