cp's OEIS Frontend

A119003 Maximal determinant of real n X n symmetric (+1,-1) matrices.

%I A119003 #18 Jun 18 2025 00:54:59
%S A119003 1,0,4,16,48,160,576,4096,14336,65536
%N A119003 Maximal determinant of real n X n symmetric (+1,-1) matrices.
%C A119003 Computation of the determinant of these two matrices:
%C A119003   {-1, -1, -1, -1,  1,  1,  1, -1},
%C A119003   {-1,  1, -1,  1,  1,  1, -1,  1},
%C A119003   {-1, -1,  1,  1,  1, -1, -1, -1},
%C A119003   {-1,  1,  1,  1, -1,  1,  1, -1},
%C A119003   { 1,  1,  1, -1,  1,  1, -1, -1},
%C A119003   { 1,  1, -1,  1,  1, -1,  1, -1},
%C A119003   { 1, -1, -1,  1, -1,  1, -1, -1},
%C A119003   {-1,  1, -1, -1, -1, -1, -1, -1}
%C A119003 and
%C A119003   {-1,  1,  1, -1,  1, -1,  1,  1,  1},
%C A119003   { 1, -1,  1, -1,  1,  1,  1,  1, -1},
%C A119003   { 1,  1,  1,  1,  1, -1, -1,  1, -1},
%C A119003   {-1, -1,  1,  1, -1,  1,  1, -1,  1},
%C A119003   { 1,  1,  1, -1, -1, -1,  1, -1, -1},
%C A119003   {-1,  1, -1,  1, -1,  1,  1,  1, -1},
%C A119003   { 1,  1, -1,  1,  1,  1,  1, -1,  1},
%C A119003   { 1,  1,  1, -1, -1,  1, -1,  1,  1},
%C A119003   { 1, -1, -1,  1, -1, -1,  1,  1,  1}
%C A119003 shows that a(8) = A003433(8) = 4096 and a(9) = A003433(9) = 14336. - _Jean-François Alcover_, Nov 19 2017
%C A119003 a(n) = n^(n/2) once there exists a symmetric Hadamard matrix of order n. In particular, a(12) = 12^6, a(16) = 16^8, etc. - _Max Alekseyev_, Jun 17 2025
%H A119003 <a href="/index/De#determinants">Index entries for sequences related to maximal determinants</a>
%H A119003 N. A. Balonin, Y. N. Balonin, D. Z. Djokovic, D. A. Karbovskiy, M. B. Sergeev. <a href="https://arxiv.org/abs/1708.05098">Construction of symmetric Hadamard matrices</a>, arXiv:1708.05098 [math.CO], 2017.
%H A119003 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hadamard%27s_maximal_determinant_problem">Hadamard's maximal determinant problem</a>.
%Y A119003 Cf. A003433, A119002, A119001, A119005.
%K A119003 nonn,more
%O A119003 1,3
%A A119003 _Giovanni Resta_, May 08 2006
%E A119003 a(8) and a(9) from _Jean-François Alcover_, Nov 19 2017
%E A119003 a(10) from _Max Alekseyev_, Jun 17 2025