login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Maximal excess of a Hadamard matrix of order 4n.
(Formerly M4489)
3

%I M4489 #15 Jan 31 2022 01:27:26

%S 0,8,20,36,64,80,112,140,172,216,244,280,324,364,408

%N Maximal excess of a Hadamard matrix of order 4n.

%C This is the maximal value of the sum of the entries of any n X n Hadamard matrix (cf. A019442).

%D Brown, Thomas A. and Spencer, Joel H., Minimization of +-1 matrices under line shifts. Colloq. Math. 23 (1971), 165-171, 177 (errata).

%D 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]

%D S. Kounias and N. Farmakis, On the excess of Hadamard matrices, Discrete Mathematics, 68 (1988), 59-69. [From _William P. Orrick_, Mar 26 2009]

%D 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.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H M. R. Best, <a href="http://dx.doi.org/10.1016/1385-7258(77)90049-X">The excess of a Hadamard matrix</a>, Indagat. Mathem. (Proceedings) 80 (1977), no. 5., 357-361

%H <a href="/index/Ha#Hadamard">Index entries for sequences related to Hadamard matrices</a>

%F n^2*2^(-n)*binomial(n,n/2) <= a(n) <= n*sqrt(n).

%F Contribution from _William P. Orrick_, Mar 26 2009: (Start)

%F 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,

%F a(n/4) <= 8[nm/4+1/2[n/4(n/4-1)/(2m)]-(n+4)/8]+n+4, if 4m^2+2m+1<n/4<=4m^2+4m+1,

%F 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.

%F [x] denotes the integer part. (See Kounias and Farmakis, 1988.) (End)

%Y Cf. A019442.

%K nonn,hard,more,nice

%O 0,2

%A _N. J. A. Sloane_

%E a(7)-a(14) from _William P. Orrick_, Mar 26 2009