The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A325588 Number of necklace compositions of n with equal circular differences up to sign. 4
 1, 2, 3, 4, 4, 7, 5, 9, 8, 10, 8, 17, 9, 14, 15, 22, 12, 23, 14, 31, 23, 25, 19, 48, 25, 35, 36, 56, 33, 59, 43, 86, 64, 74, 76, 136, 95, 127, 138, 219, 178, 245, 249, 372, 370, 445, 506, 747, 730, 907, 1069, 1431, 1544, 1927, 2268, 2981, 3332, 4074, 4896, 6320 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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). LINKS Andrew Howroyd, Table of n, a(n) for n = 1..200 EXAMPLE The a(1) = 1 through a(8) = 9 compositions:   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)        (11)  (12)   (13)    (14)     (15)      (16)       (17)              (111)  (22)    (23)     (24)      (25)       (26)                     (1111)  (11111)  (33)      (34)       (35)                                      (222)     (1111111)  (44)                                      (1212)               (1232)                                      (111111)             (1313)                                                           (2222)                                                           (11111111) MATHEMATICA neckQ[q_]:=Array[OrderedQ[{q, RotateRight[q, #]}]&, Length[q]-1, 1, And]; Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], neckQ[#]&&SameQ@@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])) )} w(n, s)={sum(k=1, n, my(R=matrix(n, n, i, j, i==j&&abs(i-k)==s), t=0, m=1); while(R, R=step(R, n, s); m++; t+=sumdiv(n, d, R[d, k]*d*eulerphi(n/d))/m ); t/n)} a(n) = {numdiv(max(1, n)) + sum(s=1, n-1, w(n, s))} \\ Andrew Howroyd, Aug 24 2019 CROSSREFS Cf. A000079, A000740, A008965, A049988, A175342, A325549, A325556, A325558, A325590. Sequence in context: A347700 A049988 A079247 * A244903 A342337 A167932 Adjacent sequences:  A325585 A325586 A325587 * A325589 A325590 A325591 KEYWORD nonn AUTHOR Gus Wiseman, May 11 2019 EXTENSIONS Terms a(26) and beyond from Andrew Howroyd, Aug 24 2019 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 25 03:56 EDT 2021. Contains 348237 sequences. (Running on oeis4.)