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A145692 Number of distinct vertex-magic 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. 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

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

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Last modified January 17 15:12 EST 2020. Contains 330958 sequences. (Running on oeis4.)