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%I #11 Jun 19 2021 22:29:05
%S 1,2,3,4,4,6,7,9,13,12,17,21,28,26,49,46,74,68,113,107,176,144,255,
%T 235,375
%N Number of necklace compositions of n such that every distinct circular subsequence has a different sum.
%C 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.
%C A circular subsequence is a sequence of consecutive terms where the first and last parts are also considered consecutive.
%e The a(1) = 1 through a(8) = 13 necklace compositions:
%e (1) (2) (3) (4) (5) (6) (7) (8)
%e (11) (12) (13) (14) (15) (16) (17)
%e (111) (22) (23) (24) (25) (26)
%e (1111) (11111) (33) (34) (35)
%e (222) (124) (44)
%e (111111) (142) (125)
%e (1111111) (152)
%e (2222)
%e (11111111)
%t neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
%t subalt[q_]:=Union[ReplaceList[q,{___,s__,___}:>{s}],DeleteCases[ReplaceList[q,{t___,__,u___}:>{u,t}],{}]];
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&UnsameQ@@Total/@subalt[#]&]],{n,20}]
%Y Cf. A000079, A000740, A008965, A059966, A108917, A143823, A169942, A276024.
%Y Cf. A325676, A325680, A325685, A325687.
%K nonn,more
%O 1,2
%A _Gus Wiseman_, May 13 2019
%E a(21)-a(25) from _Robert Price_, Jun 19 2021