A295771 -id:A295771 - cp's OEIS frontend

A295774 a(n) is the minimum size of a restricted planar additive basis for the square [0,2n]^2.

1, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92

Offset: 0

Jukka Kohonen, Nov 27 2017

A planar additive basis for the square [0,2n]^2 is a set of points with nonnegative integer coordinates such that their pairwise sums cover the square. Self-sums are allowed. In a restricted basis all points are in the lower left quadrant [0,n]^2.

Matches A008574 at least up to a(23)=92.

			a(2)=8: the square [0,4]^2 is covered by the pairwise sums of {(0,0),(1,0),(2,0),(0,1),(2,1),(0,2),(1,2),(2,2)}, which has 8 elements.

J. Kohonen, V. Koivunen and R. Rajamäki, Planar additive bases for rectangles, arXiv preprint arXiv:1711.08812, [math.NT], 2017.
J. Kohonen, V. Koivunen and R. Rajamäki, Planar additive bases for rectangles, Journal of Integer Sequences, 21 (2018), Article 18.9.8.

A295771 is the nonrestricted version.

A306608 Table read by antidiagonals: T(x,y) is the minimum size of a planar additive basis for the rectangle [0,x]*[0,y], for x,y >= 0.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 3, 4, 4, 3, 3, 5, 4, 5, 3, 4, 5, 6, 6, 5, 4, 4, 6, 6, 7, 6, 6, 4, 4, 6, 7, 8, 8, 7, 6, 4, 4, 7, 8, 9, 8, 9, 8, 7, 4, 5, 7, 8, 9, 10, 10, 9, 8, 7, 5, 5, 8, 8, 10, 10, 11, 10, 10, 8, 8, 5, 5, 8, 10, 11, 11, 12, 12, 11, 11, 10, 8, 5

Offset: 0

Jukka Kohonen, Feb 28 2019

A planar additive basis is a set of points with nonnegative integer coordinates such that their pairwise sums cover a given rectangle of points with integer coordinates. Pairwise sums of a point with itself are included.

T(x,y) = T(y,x).

			The table starts:
  1, 2, 2, 3, 3, 4, 4, ...
  2, 3, 4, 5, 5, 6, ...
  2, 4, 4, 6, 6, ...
  3, 5, 6, 7, ...
  3, 5, 6, ...
  4, 6, ...
  4, ...
  ...
T(6,3)=9: The rectangle [0,6]*[0,3] has the following minimum basis of 9 elements, with elements marked as "*", and empty locations as "-".
  3  *------
  2  ---*---
  1  **-*---
  0  ***--*-
     0123456

J. Kohonen, V. Koivunen and R. Rajamäki, Planar additive bases for rectangles, Journal of Integer Sequences, 21 (2018), Article 18.9.8. [see Table 2]

Main diagonal is A295771.

cp's OEIS Frontend

A295774 a(n) is the minimum size of a restricted planar additive basis for the square [0,2n]^2.

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