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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.

%I A209032 #10 Mar 13 2017 04:25:13
%S A209032 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 A209032 38,88,144,86,13,1,5,12,60,187,471,576,278,21,1,5,15,88,358,1237,2517,
%U A209032 2613,873,45,1,6,19,125,625,2798,8235,14611,11841,2938,83,1,6,22,170,1023
%N 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.
%C A209032 Table starts
%C A209032 ..1...1....1.....1.....1......1......1.......1.......1.......1.......1........1
%C A209032 ..1...2....2.....3.....3......4......4.......5.......5.......6.......6........7
%C A209032 ..1...2....4.....5.....7.....10.....12......15......19......22......26.......31
%C A209032 ..2...6...12....23....38.....60.....88.....125.....170.....226.....292......371
%C A209032 ..2..11...34....88...187....358....625....1023....1584....2355....3374.....4700
%C A209032 ..4..33..144...471..1237...2798...5648...10483...18174...29863...46918....71037
%C A209032 ..6..86..576..2517..8235..22249..52208..110285..214440..390344..672932..1108883
%C A209032 .13.278.2613.14611.58524.186765.505857.1210780.2631514.5293759.9995616.17902216
%H A209032 R. H. Hardin, <a href="/A209032/b209032.txt">Table of n, a(n) for n = 1..184</a>
%F A209032 Empirical for row n:
%F A209032 n=2: a(k) = a(k-1) + a(k-2) - a(k-3).
%F A209032 n=3: a(k) = 2*a(k-1) - a(k-2) + a(k-3) - 2*a(k-4) + a(k-5).
%F A209032 n=4: a(k) = 3*a(k-1) - 2*a(k-2) - 2*a(k-3) + 3*a(k-4) - a(k-5).
%F A209032 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 A209032 Some solutions for n=6, k=6:
%e A209032 .-4...-3...-2...-4...-2...-5...-3...-2...-5...-6...-2...-3...-3...-3...-4...-3
%e A209032 .-2....1...-1....2...-1...-5...-1...-1...-1...-2...-1...-1...-3...-3...-4....1
%e A209032 ..2...-2...-1...-3....0...-1...-1....0....5....3....0....3...-3...-1...-1...-2
%e A209032 .-1....1....2....0...-1....5....1....3....2....5...-1...-1....1....2....4....3
%e A209032 ..3....1....3....3....0....6....5...-1...-1....0....4....3....5....4....4....0
%e A209032 ..2....2...-1....2....4....0...-1....1....0....0....0...-1....3....1....1....1
%Y A209032 Row 2 is A004526(n+2).
%Y A209032 Row 3 is A007997(n+5).
%Y A209032 Row 4 is A084570.
%K A209032 nonn,tabl
%O A209032 1,5
%A A209032 _R. H. Hardin_, Mar 04 2012