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 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. 12

%I

%S 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,

%T 38,88,144,86,13,1,5,12,60,187,471,576,278,21,1,5,15,88,358,1237,2517,

%U 2613,873,45,1,6,19,125,625,2798,8235,14611,11841,2938,83,1,6,22,170,1023

%N 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.

%C Table starts

%C ..1...1....1.....1.....1......1......1.......1.......1.......1.......1........1

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

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

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

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

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

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

%C .13.278.2613.14611.58524.186765.505857.1210780.2631514.5293759.9995616.17902216

%H R. H. Hardin, <a href="/A209032/b209032.txt">Table of n, a(n) for n = 1..184</a>

%F Empirical for row n:

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

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

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

%F 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).

%e Some solutions for n=6, k=6:

%e .-4...-3...-2...-4...-2...-5...-3...-2...-5...-6...-2...-3...-3...-3...-4...-3

%e .-2....1...-1....2...-1...-5...-1...-1...-1...-2...-1...-1...-3...-3...-4....1

%e ..2...-2...-1...-3....0...-1...-1....0....5....3....0....3...-3...-1...-1...-2

%e .-1....1....2....0...-1....5....1....3....2....5...-1...-1....1....2....4....3

%e ..3....1....3....3....0....6....5...-1...-1....0....4....3....5....4....4....0

%e ..2....2...-1....2....4....0...-1....1....0....0....0...-1....3....1....1....1

%Y Row 2 is A004526(n+2).

%Y Row 3 is A007997(n+5).

%Y Row 4 is A084570.

%K nonn,tabl

%O 1,5

%A _R. H. Hardin_, Mar 04 2012

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Last modified September 17 05:34 EDT 2021. Contains 347478 sequences. (Running on oeis4.)