A395909 Triangle read by rows: row n gives marks on lexicographically earliest n-mark optimal ruler.

0, 0, 1, 0, 1, 3, 0, 1, 4, 6, 0, 1, 2, 6, 9, 0, 1, 2, 6, 10, 13, 0, 1, 2, 3, 8, 13, 17, 0, 1, 2, 11, 15, 18, 21, 23, 0, 1, 2, 14, 18, 21, 24, 27, 29, 0, 1, 3, 6, 13, 20, 27, 31, 35, 36, 0, 1, 3, 6, 13, 20, 27, 34, 38, 42, 43, 0, 1, 2, 3, 23, 28, 32, 36, 40, 44, 47, 50

Offset

1,6

Comments

A ruler is optimal if it is perfect and no perfect ruler with the same number of segments has a greater length. Optimal rulers are rare. There are 155050 perfect rulers with length in [0, 123], but only 74 optimal rulers in this range. No blueprint for these rulers is known. It has been conjectured by the author that an optimal ruler with more than 13 segments is either a Wichmann ruler or the mirror image of a Wichmann ruler. For more definitions and references see A103294; more optimal rulers can be found under the Luschny link.

Links

Table of n, a(n) for n=1..78.

Peter Luschny, Perfect and Optimal Rulers

Peter Luschny, Are optimal rulers of Wichmann type?

Peter Luschny, Perfect rulers, 2011.

B. Wichmann, A note on restricted difference bases, J. Lond. Math. Soc. 38 (1963), 465-466.

Wikipedia, Sparse ruler.

Index entries for sequences related to perfect rulers

Example

Triangle starts:
  0;
  0, 1;
  0, 1, 3;
  0, 1, 4,  6;
  0, 1, 2,  6,  9;
  0, 1, 2,  6, 10, 13;
  0, 1, 2,  3,  8, 13, 17;
  0, 1, 2, 11, 15, 18, 21, 23;
  0, 1, 2, 14, 18, 21, 24, 27, 29;
  0, 1, 3,  6, 13, 20, 27, 31, 35, 36;
  0, 1, 3,  6, 13, 20, 27, 34, 38, 42, 43;
  0, 1, 2,  3, 23, 28, 32, 36, 40, 44, 47, 50;

Crossrefs

Cf. A103294, A106683, A004137, A326499.

Sequence in context: A250104 A220421 A352493 * A106683 A139601 A213191

Adjacent sequences: A395906 A395907 A395908 * A395917 A395918 A395919

Keyword

nonn,tabl

Author

Peter Luschny, May 09 2026

Status

approved