A123067 Theta series of the "Little Methuselah" quadratic form x^2 + 2*y^2 + y*z + 4*z^2.

1, 2, 2, 4, 4, 6, 8, 2, 10, 10, 2, 12, 4, 4, 10, 4, 10, 12, 10, 10, 20, 12, 4, 12, 12, 12, 8, 8, 8, 24, 8, 0, 24, 12, 8, 10, 24, 12, 6, 12, 14, 38, 8, 8, 32, 18, 8, 4, 8, 20, 20, 16, 12, 16, 24, 4, 30, 28, 4, 14, 20, 12, 2, 18, 18, 44, 20, 8, 28, 12, 10, 26, 30, 12, 28, 16, 20, 20, 8, 16, 34

Offset

0,2

Comments

The Little Methuselah form represents every integer from 1 to 30, but fails to represent 31. Every integer-valued positive definite ternary form not equivalent to it fails to represent some integer between 1 and 30. [Conway]

References

J. H. Conway, The Sensual (Quadratic) Form, M.A.A., 1997, pp. 81-82.

Links

Table of n, a(n) for n=0..80.

Example

1 + 2*x + 2*x^2 + 4*x^3 + 4*x^4 + 6*x^5 + 8*x^6 + 2*x^7 + 10*x^8 + 10*x^9 + 2*x^10 + ...

Prog

(Magma) L:=LatticeWithGram(3, [2, 0, 0, 0, 4, 1, 0, 1, 8] ); T<q> := ThetaSeries(L, 500); T;
(PARI) {a(n)= if(n<1, n==0, qfrep([2, 0, 0; 0, 4, 1; 0, 1, 8], n, 1)[n]*2)} /* Michael Somos, Oct 23 2006 */

Crossrefs

Cf. A123063, A123064, A123065, A123068.

Sequence in context: A027188 A363213 A089076 * A218897 A218064 A352227

Adjacent sequences: A123064 A123065 A123066 * A123068 A123069 A123070

Keyword

nonn

Author

N. J. A. Sloane, Sep 27 2006, corrected Oct 26 2006

Status

approved