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%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
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