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A188430
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a(n) is the maximum of the largest elements of all n-full sets, or 0 if no such set exists.
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3
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1, 0, 2, 0, 0, 3, 4, 0, 0, 4, 5, 6, 7, 7, 8, 6, 7, 8, 9, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38, 38
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OFFSET
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1,3
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COMMENTS
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Let A be a set of positive integers. We say that A is n-full if (sum A)=[n] for a positive integer n, where (sum A) is the set of all positive integers which are a sum of distinct elements of A and [n]={1,2,...,n}. The number a(n) denotes the maximum of the set {max A: (sum A)=[n]}, or 0 if there is no n-full set.
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LINKS
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L. Naranjani and M. Mirzavaziri, Full Subsets of N, Journal of Integer Sequences, 14 (2011), Article 11.5.3.
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FORMULA
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a(n) = ceiling(n/2) for n >= 20.
G.f.: x*(1 - x + x^2 - x^3 - 2*x^4 + 5*x^5 + x^6 - 7*x^7 - x^8 + 8*x^9 + x^10 - 3*x^11 - x^13 - 2*x^15 + 3*x^17 - x^21) / ((1 - x)^2*(1 + x)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>22.
(End)
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MATHEMATICA
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LinearRecurrence[{1, 1, -1}, {1, 0, 2, 0, 0, 3, 4, 0, 0, 4, 5, 6, 7, 7, 8, 6, 7, 8, 9, 10, 11, 11}, 80] (* Harvey P. Dale, Jul 24 2021 *)
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PROG
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(Haskell)
a188430 n = a188430_list !! (n-1)
a188430_list = [1, 0, 2, 0, 0, 3, 4, 0, 0, 4, 5, 6, 7, 7, 8, 6, 7, 8, 9] ++
(drop 19 a008619_list)
(PARI) Vec(x*(1 - x + x^2 - x^3 - 2*x^4 + 5*x^5 + x^6 - 7*x^7 - x^8 + 8*x^9 + x^10 - 3*x^11 - x^13 - 2*x^15 + 3*x^17 - x^21) / ((1 - x)^2*(1 + x)) + O(x^80)) \\ Colin Barker, May 11 2020
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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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STATUS
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approved
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