A332546 a(n) is the maximal size of a set of equiangular lines of rank n.

1, 3, 6, 6, 10, 16, 28, 14, 18, 18, 20, 22, 26, 28, 36, 40

Offset

1,2

Comments

In the literature this sequence is denoted by M*(n), and A002853(n) is called M(n). Of course M*(n) <= M(n).

Comments from Yen-Chi Roger Lin, Feb 20 2020 (Start):

Wei-Hsuan Yu and I checked up to M*(10) = 18 in our paper.

Lemmens-Seidel (1973) implies that M_{1/3}(n) = 2*n - 2 for n >= 8. Up to n=12, no other angles whose reciprocal is an odd integer are possible because of the relative bound.

For n=11, there is no conference graph of order 22 in R^11 (see Theorem 11 of Fickus and Mixon), therefore M*(11) = M_{1/3}(11) = 20.

For n=12, M*(12) = M_{1/3}(12) = 22.

For n=13, M_(1/3)*(13) = 24, but M*(13) = 26. This follows from the existence of a real equiangular tight frame (of angle arccos 1/5) in R^13. Table 3 of the same Fickus-Mixon paper mentions it.

M*(15) = M(15) = 36 is an old result.

(End)

References

Lin, Yen-Chi Roger, and Wei-Hsuan Yu. "Equiangular lines and the Lemmens-Seidel conjecture." Discrete Mathematics 343.2 (2020): 111667.

Links

Table of n, a(n) for n=1..16.

Matthew Fickus and Dustin G. Mixon, Tables of the existence of equiangular tight frames, arXiv:1504.00253 [math.FA], 2015-2016.

P. W. H. Lemmens and J. J. Seidel, Equiangular lines, J. Algebra, 24 (1973), 494-512.

Yen-chi Roger Lin, Wei-Hsuan Yu, Equiangular lines and the Lemmens-Seidel conjecture, arXiv:1807.06249 [math.CO], 2018-2019. See M*(n).

Crossrefs

Cf. A002853.

Sequence in context: A349212 A316140 A147849 * A002853 A278807 A340620

Adjacent sequences: A332543 A332544 A332545 * A332547 A332548 A332549

Keyword

nonn,more

Author

N. J. A. Sloane, Feb 21 2020

Status

approved