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.
A circular subsequence is a sequence of consecutive terms where the first and last parts are also considered consecutive.
EXAMPLE
The a(1) = 1 through a(8) = 13 necklace 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) (124) (44)
(111111) (142) (125)
(1111111) (152)
(2222)
(11111111)
MATHEMATICA
neckQ[q_]:=Array[OrderedQ[{q, RotateRight[q, #]}]&, Length[q]-1, 1, And];
subalt[q_]:=Union[ReplaceList[q, {___, s__, ___}:>{s}], DeleteCases[ReplaceList[q, {t___, __, u___}:>{u, t}], {}]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], neckQ[#]&&UnsameQ@@Total/@subalt[#]&]], {n, 20}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, May 13 2019
EXTENSIONS
a(21)-a(25) from Robert Price, Jun 19 2021
STATUS
approved