A280343 Number of 2 X 2 matrices with all elements in {-n,..,0,..,n} with determinant = 2*permanent.

1, 25, 81, 233, 401, 585, 1009, 1289, 1681, 2377, 2913, 3353, 4497, 5033, 5793, 7097, 8065, 8761, 10721, 11513, 12961, 14873, 16113, 17065, 19873, 21225, 22689, 25465, 27585, 28793, 32561, 33865, 36113, 39177, 41121, 43481, 48801, 50361, 52529, 56201, 59793

Offset: 0

Author

Indranil Ghosh, Jan 01 2017

Keywords

• nonn

Links

• Indranil Ghosh and Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms n = 0..145 from Indranil Ghosh)

Crossrefs

Cf. A016754 (number of 2 X 2 matrices with all elements in {0,..,n} with determinant = 2*permanent).

Programs

Python

def t(n):
    s=0
    for a in range(-n,n+1):
        for b in range(-n,n+1):
            for c in range(-n,n+1):
                for d in range(-n,n+1):
                    if (a*d-b*c)==2*(a*d+b*c):
                        s+=1
    return s
for i in range(0,146):
    print(str(i)+" "+str(t(i)))

Python

def a280343_gen(lim):
    yield 1
    N = lim*lim
    M = (N+2)//3
    a = np.zeros(N, dtype=np.int64) # Counts solutions to x*y=k with x,y in 1..n
    for n in range(1, lim):
        a[n:n*n:n] += 2
        a[n*n] += 1
    yield (4*n+1)**2 + 8*int(a[3::3] @ a[1:M]) # David Radcliffe, Jun 14 2025

Formula

a(n) = (4*n+1)^2 + 8 * Sum_{k>0} d(k,n)*d(3*k,n) where d(k,n) is the number of integer solutions to x*y = k with 1 <= x,y <= n. - David Radcliffe, Jun 14 2025