A208970 T(n,k) is the number of n-bead necklaces labeled with numbers -k..k allowing reversal, with sum zero and first and second differences in -k..k.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 1, 1, 2, 2, 4, 3, 2, 1, 2, 2, 8, 9, 8, 2, 1, 2, 5, 11, 19, 29, 15, 4, 1, 3, 5, 18, 40, 90, 87, 42, 4, 1, 3, 5, 24, 77, 221, 371, 325, 94, 7, 1, 3, 8, 35, 130, 495, 1185, 1755, 1148, 246, 7, 1, 3, 8, 45, 213, 967, 3186, 6883, 8092, 4168, 613, 14, 1

Offset: 1

Author

R. H. Hardin, Mar 03 2012

Keywords

• nonn
• tabl

Comments

Table starts

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

.1..1...1....2....2.....2.....2......3......3......3.......3.......4......4

.1..1...2....2....2.....5.....5......5......8......8.......8......13.....13

.1..3...4....8...11....18....24.....35.....45.....61......76......98....119

.1..3...9...19...40....77...130....213....325....484.....687.....956...1294

.2..8..29...90..221...495...967...1801...3093...5050....7921...11994..17488

.2.15..87..371.1185..3186..7425..15658..30368..55222...95087..156612.248194

.4.42.325.1755.6883.21830.58791.140429.304536.612054.1154448.2066531

Examples

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

Links

• R. H. Hardin, Table of n, a(n) for n = 1..184

Crossrefs

Row 2 is A002265(n+4).

Row 3 is A000982(floor(n/3)+1).

Formula

Empirical for row n:

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

n=3: a(k) = a(k-1) + a(k-3) - a(k-4) + a(k-6) - a(k-7) - a(k-9) + a(k-10).

n=4: a(k) = 2*a(k-1) + a(k-2) - 4*a(k-3) + a(k-4) + 2*a(k-5) - a(k-6).