1,4

The inverse of an orthogonal matrix is its transpose. This implies the dot product of a row with itself must be 1. This further implies the number of ones in each row must be odd. Given that orthogonal matrices form a group, it must be the case the transpose is also an orthogonal matrix. This requires every column of a binary orthogonal matrix also have an odd number of ones. As a result, there will always be an orthogonal matrix of size n X n having rows with n-1 number of ones if n is an even number, namely an all-ones matrix except for zeros down the main diagonal. An n X n orthogonal matrix cannot exist with n-1 ones in each row if n is odd, since n-1 is even.

a(n) = n-1 if n is even.

a(n) < n-1 if n is odd.

There exist 10 X 10 binary orthogonal matrices such that every row has at least 9 ones, but no 10 X 10 binary orthogonal matrix exists with 10 ones in each row, so a(10) = 9.

There exist 9 X 9 binary orthogonal matrices such that every row has at least 5 ones, but no 9 X 9 binary orthogonal matrix exists with 6 or more ones in each row, so a(9) = 5.

Cf. A003053.

Sequence in context: A318661 A089654 A233526 * A097062 A350948 A324894

Adjacent sequences: A344671 A344672 A344673 * A344675 A344676 A344677

nonn,hard,more

Nathan J. Russell, May 26 2021

a(11)-a(12) from Martin Ehrenstein, Jun 17 2021

approved