A094384 Determinant of n X n partial Hadamard matrix with coefficient m(i,j) 1<=i,j<=n (see comment).

1, -2, 4, 16, -32, -128, -512, 4096, -8192, -32768, -131072, 1048576, 4194304, -33554432, 268435456, 4294967296, -8589934592, -34359738368, -137438953472, 1099511627776, 4398046511104, -35184372088832, 281474976710656

Offset

1,2

Comments

Let M(infinity) be the infinite matrix with coefficient m(i,j) i>=1, j>=1 defined as follows : M(0)=1 and M(k) is the 2^k X 2^k matrix following the recursion : +M(k-1)-M(k-1) M(k)= -M(k-1)-M(k-1).

Links

Table of n, a(n) for n=1..23.

Sela Fried, On sequence A094384, 2026.

Formula

It appears that abs(a(n)) = 2^A000788(n).

What is the rule for signs? Does Sum_{k=1..n} a(k+1)/a(k) = 0 iff n is in A073536?.

Conjecture: a(n) = (-2)^A000788(n-1). - Chai Wah Wu, Nov 12 2024

The comment above regarding the signs is false but the conjecture is true (see Fried link). - Sela Fried, Feb 05 2026

Example

M(2) = [1,-1;-1,-1] then a(2) = detM(2) = -2.

Prog

(Python)
from sympy import Matrix
def A094384(n):
    m = Matrix([1])
    for i in range((n-1).bit_length()):
        m = Matrix([[m, -m], [-m, -m]])
    return m[:n, :n].det() # Chai Wah Wu, Nov 12 2024

Crossrefs

Cf. A000788, A073536.

Sequence in context: A145119 A081411 A269758 * A053038 A001088 A101926

Adjacent sequences: A094381 A094382 A094383 * A094385 A094386 A094387

Keyword

sign

Author

Benoit Cloitre, Jun 03 2004

Status

approved