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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.

%I A208970 #10 Mar 13 2017 04:24:53
%S A208970 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,
%T A208970 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,
%U A208970 130,495,1185,1755,1148,246,7,1,3,8,45,213,967,3186,6883,8092,4168,613,14,1
%N 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.
%C A208970 Table starts
%C A208970 .1..1...1....1....1.....1.....1......1......1......1.......1.......1......1
%C A208970 .1..1...1....2....2.....2.....2......3......3......3.......3.......4......4
%C A208970 .1..1...2....2....2.....5.....5......5......8......8.......8......13.....13
%C A208970 .1..3...4....8...11....18....24.....35.....45.....61......76......98....119
%C A208970 .1..3...9...19...40....77...130....213....325....484.....687.....956...1294
%C A208970 .2..8..29...90..221...495...967...1801...3093...5050....7921...11994..17488
%C A208970 .2.15..87..371.1185..3186..7425..15658..30368..55222...95087..156612.248194
%C A208970 .4.42.325.1755.6883.21830.58791.140429.304536.612054.1154448.2066531
%H A208970 R. H. Hardin, <a href="/A208970/b208970.txt">Table of n, a(n) for n = 1..184</a>
%F A208970 Empirical for row n:
%F A208970 n=2: a(k) = a(k-1) + a(k-4) - a(k-5).
%F A208970 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).
%F A208970 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).
%e A208970 Some solutions for n=5, k=5:
%e A208970 .-2...-2...-1...-3...-2...-1...-2...-2...-2...-1...-2...-1...-3...-1...-2...-2
%e A208970 .-2...-1....0...-2...-1...-1...-1....0....0....0....0....0...-1....0....0...-2
%e A208970 ..0....1....1....2....2....0....2...-1....2...-1....1....0....2....0....0...-1
%e A208970 ..2....2...-1....2....1....2....0....2...-1....0....1....0....3....1....2....3
%e A208970 ..2....0....1....1....0....0....1....1....1....2....0....1...-1....0....0....2
%Y A208970 Row 2 is A002265(n+4).
%Y A208970 Row 3 is A000982(floor(n/3)+1).
%K A208970 nonn,tabl
%O A208970 1,12
%A A208970 _R. H. Hardin_, Mar 03 2012