A240117 Schoenheim lower bound L(n,6,2).

1, 3, 3, 3, 4, 4, 6, 7, 7, 8, 8, 12, 12, 13, 14, 14, 19, 20, 20, 21, 22, 27, 28, 29, 30, 31, 38, 39, 40, 41, 42, 50, 51, 52, 54, 55, 63, 65, 66, 68, 69, 79, 80, 82, 84, 85, 96, 98, 99, 101, 103, 114, 116, 118, 120, 122, 135, 137, 139, 141, 143, 157, 159, 161

Offset: 6

Author

Colin Barker, Apr 01 2014

Keywords

• nonn

Links

• Colin Barker, Table of n, a(n) for n = 6..1000
• D. Gordon, G. Kuperberg and O. Patashnik, New constructions for covering designs, arXiv:math/9502238 [math.CO], 1995.

Crossrefs

Cf. A240115, A240116, A240118, A240119.

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

Programs

Mathematica

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];
Table[schoenheim[n, 6, 2], {n, 6, 100}] (* Jean-François Alcover, Jan 26 2019, from PARI *)

PARI

schoenheim(n, k, t) = {
  my(lb = 1);
  n += 1-t; k += 1-t;
  while(t>0,
    lb = ceil((lb*n)/k);
    t--; n++; k++
  );
  lb
}
s=[]; for(n=6, 100, s=concat(s, schoenheim(n, 6, 2))); s

Formula

Empirical g.f.: x^6*(x^35 -x^31 -x^30 +2*x^26 +x^23 +x^20 -x^18 +x^17 +x^16 +x^13 -x^12 +2*x^11 +x^7 -x^5 +x^4 +2*x +1) / ( -x^36 +x^35 +x^31 -x^30 +x^6 -x^5 -x +1).