A006376 Number of indecomposable positive definite ternary quadratic forms of determinant n. (Formerly M0402)

0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 2, 2, 0, 0, 3, 1, 1, 2, 3, 1, 2, 1, 3, 2, 1, 2, 5, 1, 1, 3, 5, 2, 3, 1, 5, 5, 2, 2, 6, 2, 2, 5, 6, 3, 3, 3, 8, 4, 3, 2, 8, 4, 4, 4, 7, 5, 6, 3, 9, 6, 3, 5, 9, 4, 4, 8, 9, 4, 5, 3, 11, 9, 4, 5, 11, 5, 4, 6, 13, 5, 8, 6, 12, 8, 5, 5, 13, 6, 7, 9, 13, 9, 6

Offset

1,12

Comments

Following the conventions by Conway and Sloane, a (classically) integral ternary quadratic form is given by F(x,y,z) = a*x^2 + b*y^2 + c*z^2 + 2*d*y*z + 2*e*x*z + 2*f*x*y for some integers a,b,c,d,e,f. F is positive definite if F(x,y,z) > 0 for all integers (x,y,z) != (0,0,0). F is indecomposable if it cannot be written as the orthogonal sum of two (classically integral) positive definite quadratic forms. Here, the determinant D of F is the determinant of the corresponding 3 x 3 symmetric matrix [[a,f,e],[f,b,d],[e,d,c]], i.e. D = a*b*c + 2*d*e*f - a*d^2 - b*e^2 - c*f^2. - Robin Visser, Jun 02 2025

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 398.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Robin Visser, Table of n, a(n) for n = 1..250

G. Eisenstein, Tabelle der reducirten positiven ternären quadratischen Formen, nebst den Resultaten neuer Forschungen über diese Formen, in besonderer Rücksicht auf ihre tabellarische Berechnung, J. Reine Angew. Math. 41 (1851), 141-190 and 227-242.

A. Schiemann, Brandt-Intrau tables recomputed

Example

From Robin Visser, Jun 08 2025: (Start)
If n < 4, then there are no indecomposable positive definite ternary quadratic forms of determinant n, thus a(n) = 0 for all n < 4.
For n = 4, there is exactly 1 indecomposable positive definite ternary quadratic form of determinant 4 (up to isometry), given by 2*x^2 + 2*y^2 + 2*z^2 + 2*x*z + 2*y*z, thus a(4) = 1.
For n = 7, there is exactly 1 indecomposable positive definite ternary quadratic form of determinant 7 (up to isometry), given by 2*x^2 + 2*y^2 + 3*z^2 + 2*x*y + 2*y*z, thus a(7) = 1.
For n = 8, there is exactly 1 indecomposable positive definite ternary quadratic form of determinant 8 (up to isometry), given by 2*x^2 + 2*y^2 + 3*z^2 + 2*x*z + 2*y*z, thus a(8) = 1.
For n = 10, there is exactly 1 indecomposable positive definite ternary quadratic form of determinant 10 (up to isometry), given by 2*x^2 + 2*y^2 + 4*z^2 + 2*x*y + 2*y*z, thus a(10) = 1.
For n = 12, there are 2 indecomposable positive definite ternary quadratic forms of determinant 12 (up to isometry), given by 2*x^2 + 2*y^2 + 4*z^2 + 2*x*z + 2*y*z and 2*x^2 + 3*y^2 + 3*z^2 + 2*x*y + 2*x*z, thus a(12) = 2. (End)

Crossrefs

Cf. A006377, A071136.

Sequence in context: A145264 A300333 A357019 * A352562 A362326 A352909

Adjacent sequences: A006373 A006374 A006375 * A006377 A006378 A006379

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

Terms a(12), a(13), a(16) corrected and more terms from Robin Visser, Jun 02 2025

Status

approved