

A145692


Number of distinct vertexmagic total labelings on cycle C_n.


1



4, 6, 6, 20, 118, 282, 1540, 7092, 36128, 206848, 1439500, 10066876, 74931690, 613296028, 5263250382, 47965088850
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

3,1


LINKS

Table of n, a(n) for n=3..18.
Andrew Baker and Joe Sawada, Magic Labelings on Cycles and Wheels, Lecture Notes in Computer Science 5165 (Combinatorial Optimization and Applications. Second International Conference, COCOA 2008). pp. 361373.
Mukkai S. Krishnamoorthy, Allen Lavoie, Ali Dasdan, Bharath Santosh, Number of unique Edgemagic total labelings on Path P_n, arXiv:1402.2878 [math.CO], 2014.


EXAMPLE

From Gheorghe Coserea, May 23 2018: (Start)
For n=4 the a(4)=6 solutions are:
[1, 4, 8, 3, 2, 6, 5, 7]
[1, 5, 6, 4, 2, 7, 3, 8]
[1, 5, 8, 2, 4, 3, 7, 6]
[1, 7, 5, 2, 6, 3, 4, 8]
[3, 4, 8, 1, 6, 2, 7, 5]
[3, 6, 5, 1, 8, 2, 4, 7]
The solution [1, 4, 8, 3, 2, 6, 5, 7] is an encoding of the following vertexmagic labeling on C_4:
7 1 4 8 3 2 6
oooo
 
++
5
In this labeling vertices are labeled 7, 4, 3, 6 while edges are labeled 1, 8, 2, 5 respectively. The vertexmagic constant of labeling k is 13 since k = 5+7+1 = 1+4+8 = 8+3+2 = 2+6+5.
In general, for C_n the magic constant of labeling k satisfies 3*n+1  floor((n1)/2) <= k <= 3*n+2 + floor((n1)/2) and this bounds are tight for n>=6.
The solutions for n=4 have been generated using the MiniZinc model (e.g. $ minizinc a D"n=4;" magiccn.mzn  sort).
(End)


PROG

(MiniZinc)
% filename: magiccn.mzn : generate solution of size n
% usage: minizinc a solnsep "" searchcompletemsg "" D"n=5; " magiccn.mzn
include "globals.mzn";
int: n;
int: lo = 3*n+1  (n1) div 2;
int: hi = 3*n+2 + (n1) div 2;
array[1..2*n] of var 1..2*n: x;
var lo..hi: h;
constraint alldifferent(x);
constraint forall([h = x[2*i1] + x[2*i] + x[2*i+1]  i in 1..n1]);
constraint h = x[2*n1] + x[2*n] + x[1];
constraint forall([x[1] < x[2*i+1]  i in 1..n1]); % break rotations
constraint x[2] < x[2*n]; % break reflection symmetry
solve satisfy;
output [show(x)];
% Gheorghe Coserea, May 22 2018


CROSSREFS

Sequence in context: A077038 A053320 A019090 * A295512 A064214 A019203
Adjacent sequences: A145689 A145690 A145691 * A145693 A145694 A145695


KEYWORD

nonn,more


AUTHOR

Andrew Baker (abaker04(AT)uoguelph.ca), Oct 16 2008


STATUS

approved



