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Schoenheim lower bound L(n,4,2).
6

%I #21 Jan 10 2024 20:17:54

%S 1,3,3,4,6,7,8,11,12,13,18,19,20,26,27,29,35,37,39,46,48,50,59,61,63,

%T 73,75,78,88,91,94,105,108,111,124,127,130,144,147,151,165,169,173,

%U 188,192,196,213,217,221,239,243,248,266,271,276,295,300,305,326

%N Schoenheim lower bound L(n,4,2).

%C Only differs from A011976 when n = 7, 9, 10, or 19. - _Nathaniel Johnston_, Jan 10 2024

%H Colin Barker, <a href="/A240115/b240115.txt">Table of n, a(n) for n = 4..1000</a>

%H 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 Empirical g.f.: x^4*(x^15 -x^13 -x^12 +2*x^10 +x^7 +x^5 +2*x +1) / ( -x^16 +x^15 +x^13 -x^12 +x^4 -x^3 -x +1).

%F a(n) = ceiling((n/4)*ceiling((n-1)/3)). - _Nathaniel Johnston_, Jan 10 2024

%t 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 Table[schoenheim[n, 4, 2], {n, 4, 100}] (* _Jean-François Alcover_, Jan 26 2019, from PARI *)

%o (PARI) schoenheim(n, k, t) = {

%o my(lb = 1);

%o n += 1-t; k += 1-t;

%o while(t>0,

%o lb = ceil((lb*n)/k);

%o t--; n++; k++

%o );

%o lb

%o }

%o s=[]; for(n=4, 100, s=concat(s, schoenheim(n, 4, 2))); s

%Y Cf. A240116, A240117, A240118, A240119.

%Y Cf. A011975, A036831, A036832, A036833, A036834, A036835, A036836.

%K nonn,easy

%O 4,2

%A _Colin Barker_, Apr 01 2014