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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 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

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Last modified August 6 00:15 EDT 2024. Contains 374957 sequences. (Running on oeis4.)