A145692 Number of distinct vertex-magic total labelings on cycle C_n.

4, 6, 6, 20, 118, 282, 1540, 7092, 36128, 206848, 1439500, 10066876, 74931690, 613296028, 5263250382, 47965088850

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. 361-373.

Mukkai S. Krishnamoorthy, Allen Lavoie, Ali Dasdan, Bharath Santosh, Number of unique Edge-magic 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 vertex-magic labeling on C_4:
     7  1  4  8  3  2  6
     o-----o-----o-----o
     |                 |
     +-----------------+
              5
In this labeling vertices are labeled 7, 4, 3, 6 while edges are labeled 1, 8, 2, 5 respectively. The vertex-magic 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((n-1)/2) <= k <= 3*n+2 + floor((n-1)/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 --soln-sep "" --search-complete-msg "" -D"n=5; " magiccn.mzn
include "globals.mzn";
int: n;
int: lo = 3*n+1 - (n-1) div 2;
int: hi = 3*n+2 + (n-1) div 2;
array[1..2*n] of var 1..2*n: x;
var lo..hi: h;
constraint alldifferent(x);
constraint forall([h = x[2*i-1] + x[2*i] + x[2*i+1] | i in 1..n-1]);
constraint h = x[2*n-1] + x[2*n] + x[1];
constraint forall([x[1] < x[2*i+1] | i in 1..n-1]); % break rotations
constraint x[2] < x[2*n]; % break reflection symmetry
solve satisfy;
output [show(x)];
% Gheorghe Coserea, May 22 2018

Crossrefs

Cf. A237426.

Sequence in context: A053320 A351649 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