A004118 Maximal excess of a Hadamard matrix of order 4n. (Formerly M4489)

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

Offset

0,2

Comments

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

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

Table of n, a(n) for n=0..14.

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

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<n/4<=4m^2+4m+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)

Crossrefs

Cf. A019442.

Sequence in context: A158865 A139570 A265207 * A082231 A318339 A272730

Adjacent sequences: A004115 A004116 A004117 * A004119 A004120 A004121

Keyword

nonn,hard,more,nice

Author

N. J. A. Sloane

Extensions

a(7)-a(14) from William P. Orrick, Mar 26 2009

Status

approved