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A240116 Schoenheim lower bound L(n,5,2).

%I A240116 #18 Jan 10 2024 20:17:34
%S A240116 1,3,3,4,4,6,7,8,8,12,12,13,14,18,19,20,21,27,28,29,30,37,38,40,41,48,
%T A240116 50,52,53,62,63,65,67,76,78,80,82,93,95,97,99,111,113,116,118,130,133,
%U A240116 136,138,152,154,157,160,174,177,180,183,199,202,205,208,225
%N A240116 Schoenheim lower bound L(n,5,2).
%H A240116 Colin Barker, <a href="/A240116/b240116.txt">Table of n, a(n) for n = 5..1000</a>
%H A240116 D. Gordon, G. Kuperberg and O. Patashnik, <a href="http://arxiv.org/abs/math/9502238">New constructions for covering designs</a>, arXiv:math/9502238 [math.CO], 1995.
%F A240116 Empirical g.f.: x^5*(x^24 -x^21 -x^20 +2*x^17 +x^14 +x^12 -x^10 +2*x^9 +x^6 -x^4 +x^3 +2*x +1) / ( -x^25 +x^24 +x^21 -x^20 +x^5 -x^4 -x +1).
%F A240116 a(n) = ceiling((n/5)*ceiling((n-1)/4)). - _Nathaniel Johnston_, Jan 10 2024
%t A240116 schoenheim[n_, k_, t_] := Module[{lb = 1, n1 = n, k1 = k, t1 = t}, n1 += 1 - t1; k1 += 1 - t1; While[t1 > 0, lb = Ceiling[(lb*n1)/k1]; t1--; n1++; k1++]; lb];
%t A240116 Table[schoenheim[n, 5, 2], {n, 5, 100}] (* _Jean-François Alcover_, Jan 26 2019, from PARI *)
%o A240116 (PARI) schoenheim(n, k, t) = {
%o A240116   my(lb = 1);
%o A240116   n += 1-t; k += 1-t;
%o A240116   while(t>0,
%o A240116     lb = ceil((lb*n)/k);
%o A240116     t--; n++; k++
%o A240116   );
%o A240116   lb
%o A240116 }
%o A240116 s=[]; for(n=5, 100, s=concat(s, schoenheim(n, 5, 2))); s
%Y A240116 Cf. A240115, A240117, A240118, A240119.
%Y A240116 Cf. A011975, A011977, A036831, A036832, A036833, A036834, A036835, A036836.
%K A240116 nonn,easy
%O A240116 5,2
%A A240116 _Colin Barker_, Apr 01 2014