A104307 -id:A104307 - cp's OEIS frontend

A104308 Number of perfect rulers of length n having the least possible largest difference between any adjacent marks that can occur amongst all perfect rulers of this length.

1, 1, 1, 2, 1, 1, 1, 2, 1, 7, 3, 1, 1, 3, 1, 3, 1, 1, 12, 3, 1, 1, 1, 4, 1, 6, 1, 1, 1, 22, 7, 1, 3, 1, 1, 1, 1, 15, 3, 1, 1, 1, 1, 14, 3, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 2, 1, 13, 3, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 7, 3, 10, 4, 2, 3, 1, 1, 7, 3, 26, 10, 10, 2, 1, 3, 1, 1, 1, 26, 10, 26, 2, 4, 8, 3, 1, 1, 1

Offset: 1

Hugo Pfoertner, Mar 01 2005

nonn

For nomenclature related to perfect and optimal rulers see Peter Luschny's "Perfect Rulers" web pages.

			a(11)=3 because 3 of the A103300(11)/2=15 perfect rulers of length 11 can be constructed using the shortest possible maximum segment length A104307(11)=3: [0,1,2,5,8,11], [0,1,4,6,9,11], [0,1,4,7,9,11], not counting their mirror images.

F. Schwartau, Y. Schröder, L. Wolf and J. Schoebel, Table of n, a(n) for n = 1..208 [a(212), a(213) commented out by _Georg Fischer_, Mar 25 2022]
Peter Luschny, Perfect and Optimal Rulers. A short introduction.
Hugo Pfoertner, Largest and smallest maximum differences of consecutive marks of perfect rulers.
F. Schwartau, Y. Schröder, L. Wolf and J. Schoebel, MRLA search results and source code, Nov 6 2020.
F. Schwartau, Y. Schröder, L. Wolf and J. Schoebel, Large Minimum Redundancy Linear Arrays: Systematic Search of Perfect and Optimal Rulers Exploiting Parallel Processing, IEEE Open Journal of Antennas and Propagation, 2 (2021), 79-85.
Index entries for sequences related to perfect rulers.

Cf. A104307 size of minimally required longest segment, A103294 definitions related to complete rulers.

A104310 Maximum length of perfect rulers that can be made from segments not exceeding n.

Original entry on oeis.org

2, 7, 18, 25, 32, 59, 71, 81, 103, 135

Offset: 1

Hugo Pfoertner, Mar 01 2005

We conjecture the extension a(8)=81. For nomenclature related to perfect and optimal rulers see Peter Luschny's "Perfect Rulers" web pages.

			The complete list of these rulers starts:
n.a(n)..rulers
1..2....[0,1,2]
2..7....[0,1,3,5,7]
3.18....[0,1,4,7,10,13,16]
4.25....[0,1,2,6,10,13,17,21,25]
........[0,1,2,6,10,14,17,21,25]
........[0,1,2,6,10,14,18,21,25]
........[0,1,3,7,11,15,19,23,25]
5.37....[0,1,2,3,8,13,18,23,28,33,37]
6.59....[0,1,4,10,16,22,28,34,40,46,52,54,57,59]

Peter Luschny, Perfect and Optimal Rulers. A short introduction.
Hugo Pfoertner, Largest and smallest maximum differences of consecutive marks of perfect rulers.
F. Schwartau, Y. Schröder, L. Wolf and J. Schoebel, MRLA search results and source code, Nov 6 2020.
F. Schwartau, Y. Schröder, L. Wolf and J. Schoebel, Large Minimum Redundancy Linear Arrays: Systematic Search of Perfect and Optimal Rulers Exploiting Parallel Processing, IEEE Open Journal of Antennas and Propagation, 2 (2021), 79-85.
Index entries for sequences related to perfect rulers.

Cf. A104307 Least maximum of differences between consecutive marks, A104309 minimum length of perfect rulers containing a segment of length n, A103294 definitions related to complete rulers.

Conjectured a(8) proven via exhaustive search and a(9) ... a(10) added by Fabian Schwartau, Yannic Schröder, Lars Wolf, Joerg Schoebel, Feb 23 2021

cp's OEIS Frontend

A104308 Number of perfect rulers of length n having the least possible largest difference between any adjacent marks that can occur amongst all perfect rulers of this length.

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