A303735 -id:A303735 - cp's OEIS frontend

A187103 Maximum order of an explicit Runge-Kutta method with n function evaluations in each step.

1, 2, 3, 4, 4, 5, 6, 6, 7, 7, 8

Offset: 1

Pontus von Brömssen, Mar 04 2011

a(n) <= n-3 for n >= 10 (Butcher 1985).
Observation: first 10 terms coincide with A303735. - Omar E. Pol, Oct 04 2018
The preceding observation (by Omar E. Pol) holds also for the 11th term. - Pontus von Brömssen, Apr 05 2023

J. C. Butcher, The non-existence of ten stage eighth order explicit Runge-Kutta methods, BIT 25 (1985) 521-540.
MathOverflow, What is the minimum number of stages s required for a Runge-Kutta type numerical method of given order p?, 2019.
Wikipedia, Runge-Kutta methods.

Cf. A087803, A187102, A303735.

a(n) = max{k; A187102(k)<=n}.

A348600 Triangle read by rows: T(n,k) is the number of (unlabeled) connected graphs with n nodes and metric dimension k, 0 <= k < n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 13, 6, 1, 0, 1, 62, 39, 9, 1, 0, 1, 275, 488, 77, 11, 1, 0, 1, 1710, 8116, 1145, 130, 14, 1, 0, 1, 12061, 216432, 29958, 2415, 196, 16, 1, 0, 1, 93706, 9512947, 2026922, 78265, 4434, 276, 19, 1

Offset: 1

Pontus von Brömssen, Jan 26 2022

			Triangle begins:
  n\k| 0  1     2       3       4     5    6   7  8  9
  ---+------------------------------------------------
   1 | 1
   2 | 0  1
   3 | 0  1     1
   4 | 0  1     4       1
   5 | 0  1    13       6       1
   6 | 0  1    62      39       9     1
   7 | 0  1   275     488      77    11    1
   8 | 0  1  1710    8116    1145   130   14   1
   9 | 0  1 12061  216432   29958  2415  196  16  1
  10 | 0  1 93706 9512947 2026922 78265 4434 276 19  1

Gary Chartrand, Linda Eroh, Mark A. Johnson, and Ortrud R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Applied Mathematics 105 (2000), 99-113.
Richard C. Tillquist, Rafael M. Frongillo, and Manuel E. Lladser, Getting the lay of the land in discrete space: a survey of metric dimension and its applications, arXiv:2104.07201 [math.CO], 2021.
Wikipedia, Metric dimension

Cf. A047209, A303735.

Row sums: A001349.

T(n,1) = 1 for n >= 2, because the only graphs with metric dimension 1 are the paths of positive lengths (Chartrand et al. 2000).
T(n,n-2) = A047209(n-2) = floor(5*n/2-6) for n >= 3 (follows from the complete description of graphs with n nodes and metric dimension n-2 by Chartrand et al. 2000).
T(n,n-1) = 1 for n >= 1 , because the only graph with n nodes and metric dimension n-1 is the complete graph (Chartrand et al. 2000).

cp's OEIS Frontend

A187103 Maximum order of an explicit Runge-Kutta method with n function evaluations in each step.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula