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%I #38 Jul 24 2021 10:53:09
%S 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,
%T 15,16,16,17,17,18,18,19,19,20,20,21,21,22,22,23,23,24,24,25,25,26,26,
%U 27,27,28,28,29,29,30,30,31,31,32,32,33,33,34,34,35,35,36,36,37,37,38,38
%N a(n) is the maximum of the largest elements of all n-full sets, or 0 if no such set exists.
%C 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.
%H Mohammad Saleh Dinparvar, <a href="http://github.com/SalehDinparvar/sequence_computer/blob/master/A188430.py">Python program</a> (github)
%H L. Naranjani and M. Mirzavaziri, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Mirzavaziri/mirza4.html">Full Subsets of N</a>, Journal of Integer Sequences, 14 (2011), Article 11.5.3.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F a(n) = ceiling(n/2) for n >= 20.
%F From _Colin Barker_, May 11 2020: (Start)
%F 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)).
%F a(n) = a(n-1) + a(n-2) - a(n-3) for n>22.
%F (End)
%t 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 *)
%o (Haskell)
%o a188430 n = a188430_list !! (n-1)
%o a188430_list = [1, 0, 2, 0, 0, 3, 4, 0, 0, 4, 5, 6, 7, 7, 8, 6, 7, 8, 9] ++
%o (drop 19 a008619_list)
%o -- _Reinhard Zumkeller_, Aug 06 2015
%o (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
%Y Cf. A188429, A188431.
%Y Cf. A008619.
%K nonn,easy
%O 1,3
%A _Madjid Mirzavaziri_, Mar 31 2011
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