A209485 T(n,k) is the number of n-bead necklaces labeled with numbers -k..k allowing reversal, with sum zero and avoiding the patterns z z+1 z+2 and z z-1 z-2.

1, 1, 2, 1, 3, 1, 1, 4, 4, 4, 1, 5, 7, 15, 4, 1, 6, 12, 35, 38, 11, 1, 7, 17, 72, 140, 136, 15, 1, 8, 24, 128, 390, 731, 458, 43, 1, 9, 31, 205, 866, 2606, 3740, 1781, 77, 1, 10, 40, 311, 1702, 7179, 17771, 20888, 6912, 199, 1, 11, 49, 448, 3014, 16660, 60778, 128598, 118137

Offset: 1

Author

R. H. Hardin, Mar 09 2012

Keywords

• nonn
• tabl

Comments

Table starts

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

..2....3.....4......5......6.......7.......8........9.......10.......11......12

..1....4.....7.....12.....17......24......31.......40.......49.......60......71

..4...15....35.....72....128.....205.....311......448......618......829....1083

..4...38...140....390....866....1702....3014.....4984.....7774....11620...16716

.11..136...731...2606...7179...16660...34233....64220...112263...185506..292759

.15..458..3740..17771..60778..168453..401634...857433..1679810..3074315.5321674

.43.1781.20888.128598.541494.1778878.4907310.11891268.26069478.52776268

Examples

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

Links

• R. H. Hardin, Table of n, a(n) for n = 1..148

Crossrefs

Row 3 is A074148.

Row 4 is A209345.

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-3) - 2*a(k-5) + 2*a(k-6) + a(k-8) - 2*a(k-10) + a(k-11).

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