cp's OEIS Frontend

A208777 T(n,k) is the number of n-bead necklaces labeled with numbers 1..k not allowing reversal, with no adjacent beads differing by more than 1.

%I A208777 #20 Mar 13 2017 04:24:33
%S A208777 1,2,1,3,3,1,4,5,4,1,5,7,7,6,1,6,9,10,12,8,1,7,11,13,18,19,14,1,8,13,
%T A208777 16,24,30,39,20,1,9,15,19,30,41,65,71,36,1,10,17,22,36,52,91,128,152,
%U A208777 60,1,11,19,25,42,63,117,185,293,315,108,1,12,21,28,48,74,143,242,435,658,685
%N A208777 T(n,k) is the number of n-bead necklaces labeled with numbers 1..k not allowing reversal, with no adjacent beads differing by more than 1.
%C A208777 Table starts
%C A208777 .1..2...3...4...5...6...7...8....9...10...11...12...13...14...15...16...17...18
%C A208777 .1..3...5...7...9..11..13..15...17...19...21...23...25...27...29...31...33...35
%C A208777 .1..4...7..10..13..16..19..22...25...28...31...34...37...40...43...46...49...52
%C A208777 .1..6..12..18..24..30..36..42...48...54...60...66...72...78...84...90...96..102
%C A208777 .1..8..19..30..41..52..63..74...85...96..107..118..129..140..151..162..173..184
%C A208777 .1.14..39..65..91.117.143.169..195..221..247..273..299..325..351..377..403..429
%C A208777 .1.20..71.128.185.242.299.356..413..470..527..584..641..698..755..812..869..926
%C A208777 .1.36.152.293.435.577.719.861.1003.1145.1287.1429.1571.1713.1855.1997.2139.2281
%C A208777 The transposed array (starting with index 0) appears as Table 2 in the Knopfmacher et al. reference. [_Joerg Arndt_, Aug 08 2012]
%H A208777 R. H. Hardin, <a href="/A208777/b208777.txt">Table of n, a(n) for n = 1..456</a>
%H A208777 Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger, <a href="http://arxiv.org/abs/0809.0551">Smooth words and Chebyshev polynomials</a>, arXiv:0809.0551v1 [math.CO], 2008.
%e A208777 All solutions for n=4, k=3:
%e A208777 ..2....1....2....1....2....2....2....1....3....1....1....1
%e A208777 ..3....2....2....2....2....3....2....1....3....1....2....1
%e A208777 ..2....2....3....1....2....3....2....1....3....2....3....1
%e A208777 ..3....2....3....2....2....3....3....2....3....2....2....1
%t A208777 T[n_, k_] := 1/n*Sum[DivisorSum[n, EulerPhi[#]*(1+2*Cos[i*Pi/(k+1)])^(n/#)&], {i, 1, k}] // FullSimplify; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Dec 05 2015, adapted from PARI *)
%o A208777 (PARI)
%o A208777 /* from the Knopfmacher et al. reference */
%o A208777 default(realprecision,99); /* using floats */
%o A208777 sn(n,k)=1/n*sum(i=1,k,sumdiv(n,j,eulerphi(j)*(1+2*cos(i*Pi/(k+1)))^(n/j)));
%o A208777 T(n,k)=sn(n,k);
%o A208777 matrix(22,22,n,k, round(T(n,k)) ) /* as matrix shown in comments */
%o A208777 /* _Joerg Arndt_, Aug 09 2012 */
%Y A208777 Column 2 is A000031, col. 3 is A208772, col. 4 is A208773, col. 5 is A208774, col. 6 is A208775, col. 7 is A208776.
%K A208777 nonn,tabl
%O A208777 1,2
%A A208777 _R. H. Hardin_, Mar 01 2012