A208993 T(n,k) is the number of n-bead necklaces labeled with numbers -k..k not allowing reversal, with sum zero and first differences in -k..k.

1, 1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 3, 5, 7, 3, 1, 3, 7, 16, 17, 5, 1, 4, 11, 33, 59, 51, 9, 1, 4, 15, 58, 159, 253, 155, 18, 1, 5, 19, 95, 351, 874, 1111, 508, 35, 1, 5, 25, 144, 683, 2354, 4935, 5067, 1683, 73, 1, 6, 31, 209, 1207, 5404, 16307, 28816, 23483, 5709, 151, 1, 6, 37, 290

Offset

1,5

Comments

Table starts

..1...1....1.....1......1......1.......1.......1.......1........1........1

..1...2....2.....3......3......4.......4.......5.......5........6........6

..1...3....5.....7.....11.....15......19......25......31.......37.......45

..2...7...16....33.....58.....95.....144.....209.....290......391......512

..3..17...59...159....351....683....1207....1989....3099.....4623.....6647

..5..51..253...874...2354...5404...11010...20553...35781....58971....92851

..9.155.1111..4935..16307..44209..104001..219929..428031...779487..1344353

.18.508.5067.28816.116183.371893.1008880.2416967.5255962.10577092.19976373

Links

R. H. Hardin, Table of n, a(n) for n = 1..203

Formula

Empirical for row n:

n=2: a(k) = a(k-1) + a(k-2) - a(k-3).

n=3: a(k) = 2*a(k-1) - a(k-2) + a(k-3) - 2*a(k-4) + a(k-5).

n=4: a(k) = 3*a(k-1) - 2*a(k-2) - 2*a(k-3) + 3*a(k-4) - a(k-5).

n=5: a(k) = 2*a(k-1) + a(k-2) - 4*a(k-3) + a(k-4) + 3*a(k-5) - 3*a(k-6) - a(k-7) + 4*a(k-8) - a(k-9) - 2*a(k-10) + a(k-11).

Example

Some solutions for n=6, k=6:
.-3...-4...-5...-5...-6...-5...-4...-2...-4...-5...-4...-4...-3...-6...-5...-5
..0...-3...-4...-3...-1....1....2...-2...-2....1...-1....1....3....0...-1....1
..0....3....1....3....2....5....5....1....3....1....4....3....0....4....0...-2
.-1....5....5....2....3....2...-1....2....2....1....0....4....0....2....5....0
..1....1....4....2....3...-4....0...-2...-1....2....2...-1...-2....2....2....6
..3...-2...-1....1...-1....1...-2....3....2....0...-1...-3....2...-2...-1....0

Crossrefs

Row 2 is A004526(n+2).

Sequence in context: A030496 A005794 A280860 * A328029 A201384 A238348

Adjacent sequences: A208990 A208991 A208992 * A208994 A208995 A208996

Keyword

nonn,tabl

Author

R. H. Hardin, Mar 04 2012

Status

approved