A280391 Number of 2 X 2 matrices with all elements in {0,...,n} with permanent = determinant * n.

1, 12, 25, 57, 81, 141, 169, 259, 297, 413, 441, 621, 625, 825, 873, 1079, 1089, 1403, 1369, 1739, 1729, 2021, 2025, 2507, 2433, 2859, 2905, 3301, 3249, 4029, 3721, 4509, 4305, 4793, 4989, 5551, 5329, 6027, 6025, 6807, 6561, 7917, 7225, 8357, 8121, 8677, 8649, 9843, 9481, 10889

Offset

0,2

Comments

All the values except a(1) are odd.

From Robert Israel, Jan 02 2017: (Start)

Number of solutions to (n+1)*x*y = (n-1)*z*w for x,y,z,w in [0..n].

a(n) >= (2n+1)^2, with equality if n+1 is an odd prime. (End)

Links

Robert Israel and Indranil Ghosh, Table of n, a(n) for n = 0..1600 (n = 0..200 from Indranil Ghosh)

Maple

g:= proc(r, n) if r = 0 then 2*n+1 else nops(select(t -> t <= n and r <= t*n, numtheory:-divisors(r))) fi end proc:
f:= proc(n) local c;
    if n::even then (2*n+1)^2 + add(g((n+1)*c, n)*g((n-1)*c, n), c=1..n-1)
    else (2*n+1)^2 + add(g((n+1)/2*c, n) * g((n-1)/2*c, n), c=1..2*n-1)
    fi
end proc:
map(f, [$0..100]); # Robert Israel, Jan 02 2017

Mathematica

g[r_, n_] := If[r == 0, 2n + 1, Length[Select[Divisors[r], # <= n && r <= # n&]]];
f[n_] := If[EvenQ[n], (2n + 1)^2 + Sum[g[(n + 1)c, n] g[(n - 1)c, n], {c, 1, n - 1}], (2n + 1)^2 + Sum[g[(n + 1)/2 c, n] g[(n - 1)/2 c, n], {c, 1, 2n - 1}]];
f /@ Range[0, 100] (* Jean-François Alcover, Jul 29 2020, after Robert Israel *)

Prog

(Python)
def t(n):
    s=0
    for a in range(0, n+1):
        for b in range(0, n+1):
            for c in range(0, n+1):
                for d in range(0, n+1):
                    if (a*d-b*c)*n==(a*d+b*c):
                        s+=1
    return s
for i in range(0, 201):
    print str(i)+" "+str(t(i))

Crossrefs

Cf. A280321 (Number of 2 X 2 matrices with all elements in {0,..,n} with permanent*n = determinant).

Cf. A015237 (Number of 2 X 2 matrices having all elements in {0..n} with determinant = permanent).

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

Cf. A280364 (Number of 2 X 2 matrices having all elements in {0..n} with determinant^n = permanent).

Sequence in context: A198274 A229447 A175523 * A058848 A292493 A042869

Adjacent sequences: A280388 A280389 A280390 * A280392 A280393 A280394

Keyword

nonn

Author

Indranil Ghosh, Jan 02 2017

Status

approved