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 A325590 Number of necklace compositions of n with circular differences all equal to 1 or -1. 6
 0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 3, 3, 4, 4, 5, 7, 6, 7, 10, 10, 11, 15, 16, 18, 23, 25, 32, 38, 43, 53, 64, 73, 89, 108, 131, 153, 188, 223, 272, 329, 395, 475, 583, 697, 848, 1027, 1247, 1506, 1837, 2223, 2708, 3282, 3993, 4848, 5913, 7175, 8745, 10640 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,9 COMMENTS A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations. The circular differences of a sequence c of length k are c_{i + 1} - c_i for i < k and c_1 - c_i for i = k. For example, the circular differences of (1,2,1,3) are (1,-1,2,-2). Up to rotation, a(n) is the number of ways to arrange positive integers summing to n in a circle such that adjacent parts differ by 1 or -1. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..200 EXAMPLE The first 16 terms count the following compositions:    3: (12)    5: (23)    6: (1212)    7: (34)    8: (1232)    9: (45)    9: (121212)   10: (2323)   11: (56)   11: (121232)   12: (2343)   12: (12121212)   13: (67)   13: (123232)   14: (3434)   14: (12121232)   15: (78)   15: (123432)   15: (232323)   15: (1212121212)   16: (3454)   16: (12321232)   16: (12123232) The a(21) = 7 necklace compositions:   (10,11)   (2,3,4,5,4,3)   (3,4,3,4,3,4)   (1,2,1,2,1,2,3,4,3,2)   (1,2,3,2,1,2,3,2,3,2)   (1,2,1,2,3,2,3,2,3,2)   (1,2,1,2,1,2,1,2,1,2,1,2,1,2) MATHEMATICA neckQ[q_]:=Array[OrderedQ[{q, RotateRight[q, #]}]&, Length[q]-1, 1, And]; Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], neckQ[#]&&(SameQ[1, ##]&@@Abs[Differences[Append[#, First[#]]]])&]], {n, 15}] PROG (PARI) step(R, n, s)={matrix(n, n, i, j, if(i>j, if(j>s, R[i-j, j-s]) + if(j+s<=n, R[i-j, j+s])) )} a(n)={sum(k=1, n, my(R=matrix(n, n, i, j, i==j&&abs(i-k)==1), t=0, m=1); while(R, R=step(R, n, 1); m++; t+=sumdiv(n, d, R[d, k]*d*eulerphi(n/d))/m ); t/n)} \\ Andrew Howroyd, Aug 23 2019 CROSSREFS Cf. A000079, A000740, A008965, A034297, A173258, A325556, A325588, A325589, A325591. Sequence in context: A140426 A146879 A231577 * A277210 A304777 A058762 Adjacent sequences:  A325587 A325588 A325589 * A325591 A325592 A325593 KEYWORD nonn AUTHOR Gus Wiseman, May 12 2019 EXTENSIONS a(26)-a(40) from Lars Blomberg, Jun 11 2019 Terms a(41) and beyond from Andrew Howroyd, Aug 23 2019 STATUS approved

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Last modified May 27 21:52 EDT 2020. Contains 334671 sequences. (Running on oeis4.)