A209032 T(n,k) is the number of n-bead necklaces labeled with numbers -k..k allowing reversal, with sum zero and first differences in -k..k.

1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 3, 4, 6, 2, 1, 3, 5, 12, 11, 4, 1, 4, 7, 23, 34, 33, 6, 1, 4, 10, 38, 88, 144, 86, 13, 1, 5, 12, 60, 187, 471, 576, 278, 21, 1, 5, 15, 88, 358, 1237, 2517, 2613, 873, 45, 1, 6, 19, 125, 625, 2798, 8235, 14611, 11841, 2938, 83, 1, 6, 22, 170, 1023

Offset

1,5

Comments

Table starts

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

..1...2....2.....3.....3......4......4.......5.......5.......6.......6........7

..1...2....4.....5.....7.....10.....12......15......19......22......26.......31

..2...6...12....23....38.....60.....88.....125.....170.....226.....292......371

..2..11...34....88...187....358....625....1023....1584....2355....3374.....4700

..4..33..144...471..1237...2798...5648...10483...18174...29863...46918....71037

..6..86..576..2517..8235..22249..52208..110285..214440..390344..672932..1108883

.13.278.2613.14611.58524.186765.505857.1210780.2631514.5293759.9995616.17902216

Links

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

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) - 2*a(k-3) + 2*a(k-4) - a(k-5) - 2*a(k-6) + 2*a(k-7) + a(k-8) - 2*a(k-9) + 2*a(k-10) - 2*a(k-12) + a(k-13).

Example

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

Crossrefs

Row 2 is A004526(n+2).

Row 3 is A007997(n+5).

Row 4 is A084570.

Sequence in context: A342749 A131335 A225332 * A183935 A046081 A366417

Adjacent sequences: A209029 A209030 A209031 * A209033 A209034 A209035

Keyword

nonn,tabl

Author

R. H. Hardin, Mar 04 2012

Status

approved