cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A160826 Improvement of A125852 over A053416, A053479 and A053417.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 4, 3, 0, 0, 0, 1, 0, 0, 0, 0, 2, 4, 5, 1, 3, 1, 0, 3, 2, 3, 4, 3, 4, 5, 6, 9, 4, 3, 0, 1, 0, 0, 0, 2, 4, 3, 4, 5, 10, 14, 3, 6, 0, 7, 0, 4, 5, 1, 8, 6, 0, 4, 7, 8, 6, 5, 11, 5, 9, 12, 12, 4, 0, 11, 7, 12, 0, 3, 1, 0, 1, 5, 0, 6, 2, 10, 11, 25, 17, 3, 2, 0, 9, 0, 12, 5, 0, 4, 2
Offset: 1

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Author

Hagen von Eitzen, May 27 2009

Keywords

Comments

How many more lattice points of a hexagonal lattice can be covered by placing a disk of diameter n at an optimal center instead of one of the three obvious centers (a lattice point, midpoint between two lattice points, barycenter of a fundamental triangle)?
The first difference occurs at n=9, when a diameter 9 disc around e.g. (1/2, 4*sqrt(5)) covers more lattice points than one around (0,0) or (1/2,0) or (1/2,sqrt(3)/6).
Clearly a(n) = O(n) as all "extra" points have norm approximately n^2/4 if the optimal center is chosen near (0,0). Does a(n)/n converge? Are there only finitely many n with a(n)=0?

Examples

			For diameters n=2,4,6,8 a disc around (0,0) and for n=1,3,5,7 a disc around(1/2,0) happens to be optimal (covers as many points as possible); therefore a(1)=a(2)=...=a(8)=0.
a(9) = A125852(9) - max(A053416(9),A053479(9),A053417(9)) = 77 - max(73,69,76) = 1.
		

Formula

a(n) = A125852(n) - max(A053416(n),A053479(n),A053417(n))