A004118 Maximal excess of a Hadamard matrix of order 4n.
0, 8, 20, 36, 64, 80, 112, 140, 172, 216, 244, 280, 324, 364, 408
Offset: 0
References
- Brown, Thomas A. and Spencer, Joel H., Minimization of +-1 matrices under line shifts. Colloq. Math. 23 (1971), 165-171, 177 (errata).
- N. Farmakis and S. Kounias, The excess of Hadamard matrices and optimal designs, Discrete Mathematics, 67 (1987), 165-176. [From William P. Orrick, Mar 26 2009]
- S. Kounias and N. Farmakis, On the excess of Hadamard matrices, Discrete Mathematics, 68 (1988), 59-69. [From William P. Orrick, Mar 26 2009]
- Seberry, Jennifer and Yamada, Mieko; Hadamard matrices, sequences and block designs, in Dinitz and Stinson, eds., Contemporary design theory, pp. 431-560, Wiley-Intersci. Ser. Discrete Math. Optim., Wiley, New York, 1992.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- M. R. Best, The excess of a Hadamard matrix, Indagat. Mathem. (Proceedings) 80 (1977), no. 5., 357-361
- Index entries for sequences related to Hadamard matrices
Crossrefs
Cf. A019442.
Formula
n^2*2^(-n)*binomial(n,n/2) <= a(n) <= n*sqrt(n).
Contribution from William P. Orrick, Mar 26 2009: (Start)
a(n/4) <= n(2m+1)+8[n/4(n/4-1)/(2(2m+1))], if 4m^2<=n/4<=4m^2+2m+1 or 4m^2+6m+3<=n/4<=4(m+1)^2,
a(n/4) <= 8[nm/4+1/2[n/4(n/4-1)/(2m)]-(n+4)/8]+n+4, if 4m^2+2m+1
a(n/4)<=8[nm/4+1/2[n/4(n/4-1)/(2(m+1))]+(n-4)/8]+n+4, if 4m^2+4m+1<=n/4<4m^2+6m+3.
[x] denotes the integer part. (See Kounias and Farmakis, 1988.) (End)
Extensions
a(7)-a(14) from William P. Orrick, Mar 26 2009
Comments