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A334211
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a(n) = 1 + a(n-1)*(A007018(n-1) + 1), with a(0) = 1.
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0
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OFFSET
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0,2
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COMMENTS
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The denominator for the infinum of the ratio of the optimal value for the objective functions of the integer knapsack problem with n+1 types of items with the additional restriction that only one type of items is allowed to include in the solution and without it. [Sentence not clear, needs editing. - N. J. A. Sloane, Mar 07 2021] The numerator is A007018(n+1).
The sequence a(n)/A007018(n) decreases monotonically and tends to 0.591355492056890...
The infinum for a(n)/A007018(n) is achieved on the problem maximize Sum_{j=1..n} x_j/A007018(j-1), subject to Sum_{j=1..n} x_j/(A007018(j-1)+mu_n) <= 1, where 0 <= mu_n < 1 and Sum_{j=1..n} 1/(A007018(j-1) + mu_n) = 1. In particular, mu_1 = 1, mu_2 = (sqrt(5)-1)/2 =~ 0.61803, mu_3 =~ 0.93923, mu_4 =~ 0.99855. The optimal solution vector to this problem is (1,1,...,1) and the optimal solution value is a(n+1)/A007018(n+1), whereas the approximate solution vectors are (1,0,0,...,0), (0,A007018(1),0,...,0), (0,0,A007018(2),...,0), ..., (0,0,0,...,A007018(n-1)) and the corresponding value of the objective function is 1.
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LINKS
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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