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

1,3

Comments

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

Links

R. H. Hardin, Table of n, a(n) for n = 1..165

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).

Example

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

Crossrefs

Row 3 is A074148.

Sequence in context: A099478 A133913 A209485 * A294099 A209115 A353430

Adjacent sequences: A209341 A209342 A209343 * A209345 A209346 A209347

Keyword

nonn,tabl

Author

R. H. Hardin, Mar 06 2012

Status

approved