A325680 - OEIS

%I #13 Jun 19 2021 22:28:56

%S 1,1,2,4,5,6,8,14,16,29,24,42,46,78,66,146,133,242,208,386,352,620,

%T 494,948,842,1447

%N Number of compositions of n such that every distinct circular subsequence has a different sum.

%C A composition of n is a finite sequence of positive integers summing to n.

%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) = 16 compositions:

%e   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)

%e        (11)  (12)   (13)    (14)     (15)      (16)       (17)

%e              (21)   (22)    (23)     (24)      (25)       (26)

%e              (111)  (31)    (32)     (33)      (34)       (35)

%e                     (1111)  (41)     (42)      (43)       (44)

%e                             (11111)  (51)      (52)       (53)

%e                                      (222)     (61)       (62)

%e                                      (111111)  (124)      (71)

%e                                                (142)      (125)

%e                                                (214)      (152)

%e                                                (241)      (215)

%e                                                (412)      (251)

%e                                                (421)      (512)

%e                                                (1111111)  (521)

%e                                                           (2222)

%e                                                           (11111111)

%t subalt[q_]:=Union[ReplaceList[q,{___,s__,___}:>{s}],DeleteCases[ReplaceList[q,{t___,__,u___}:>{u,t}],{}]];

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Total/@subalt[#]&]],{n,0,15}]

%Y Cf. A000079, A008965, A108917, A143823, A169942, A276024.

%Y Cf. A325545, A325676, A325682, A325685, A325687, A325688.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, May 13 2019

%E a(18)-a(25) from _Robert Price_, Jun 19 2021