login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Number of 4-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and first and second differences in -n..n.
1

%I #13 Jul 07 2018 08:51:21

%S 1,3,4,8,11,18,24,35,45,61,76,98,119,148,176,213,249,295,340,396,451,

%T 518,584,663,741,833,924,1030,1135,1256,1376,1513,1649,1803,1956,2128,

%U 2299,2490,2680,2891,3101,3333,3564,3818,4071,4348,4624,4925,5225,5551,5876

%N Number of 4-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and first and second differences in -n..n.

%C Row 4 of A208970.

%H R. H. Hardin, <a href="/A208971/b208971.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).

%F Conjectures from _Colin Barker_, Jul 07 2018: (Start)

%F G.f.: x*(1 + x - 3*x^2 + x^3 + 2*x^4 - x^5) / ((1 - x)^4*(1 + x)^2).

%F a(n) = (2*n^3 + 6*n^2 + 28*n + 48) / 48 for n even.

%F a(n) = (2*n^3 + 6*n^2 + 22*n + 18) / 48 for n odd.

%F (End)

%e All solutions for n=5:

%e -2 -1 -2 -2 -1 -1 -1 0 -2 -2 -1

%e 0 -1 0 -1 0 0 1 0 -1 -2 -1

%e 2 0 1 1 0 1 -1 0 2 2 1

%e 0 2 1 2 1 0 1 0 1 2 1

%Y Cf. A208970.

%K nonn

%O 1,2

%A _R. H. Hardin_, Mar 03 2012