A209344 T(n,k) is the number of n-bead necklaces labeled with numbers -k..k allowing reversal, with sum zero with no three beads in a row equal.
1, 1, 2, 1, 3, 1, 1, 4, 4, 4, 1, 5, 7, 15, 5, 1, 6, 12, 35, 40, 14, 1, 7, 17, 72, 145, 146, 21, 1, 8, 24, 128, 400, 770, 514, 51, 1, 9, 31, 205, 883, 2698, 4029, 2032, 102, 1, 10, 40, 311, 1724, 7358, 18646, 22739, 8076, 249, 1, 11, 49, 448, 3045, 16968, 62853, 136000
Offset: 1
Examples
Table starts: ..1....1.....1......1......1.......1.......1........1........1........1 ..2....3.....4......5......6.......7.......8........9.......10.......11 ..1....4.....7.....12.....17......24......31.......40.......49.......60 ..4...15....35.....72....128.....205.....311......448......618......829 ..5...40...145....400....883....1724....3045.....5026.....7827....11684 .14..146...770...2698...7358...16968...34720....64942...113288...186906 .21..514..4029..18646..62853..172610..409199...870122..1699831..3104474 .51.2032.22739.136000.563109.1830872.5016681.12099880.26438711.53392286 Some solutions for n=6, k=8: .-4...-4...-4...-8...-7...-6...-6...-8...-7...-8...-7...-7...-8...-8...-8...-4 .-3...-3...-3...-3....0....1....1....0...-2....0....1...-2....3...-8...-4...-4 ..5...-1...-4....4...-4...-1....1....1....8....3....0....8...-4...-4....0...-2 .-2....3...-3....1....2....8....6....4...-5....5...-6....1....0....6....7....5 .-1...-1....6....3....3...-5...-6....0....5...-4....8...-7....6....7....3....7 ..5....6....8....3....6....3....4....3....1....4....4....7....3....7....2...-2
Links
- R. H. Hardin, Table of n, a(n) for n = 1..165
Crossrefs
Row 3 is A074148.
Formula
Empirical for row n:
n=2: a(k) = 2*a(k-1) - a(k-2).
n=3: a(k) = 2*a(k-1) - 2*a(k-3) + a(k-4).
n=4: a(k) = 3*a(k-1) - 3*a(k-2) + 2*a(k-3) - 3*a(k-4) + 3*a(k-5) - a(k-6).
n=5: a(k) = 2*a(k-1) + a(k-2) - 3*a(k-3) - a(k-4) + a(k-5) + 3*a(k-6) - a(k-7) - 2*a(k-8) + a(k-9).
n=6: a(k) = 4*a(k-1) - 5*a(k-2) + a(k-3) + a(k-4) + a(k-5) + a(k-6) - 5*a(k-7) + 4*a(k-8) - a(k-9).