A382399 -id:A382399 - cp's OEIS frontend

A325679 Number of compositions of n such that every restriction to a circular subinterval has a different sum.

1, 1, 1, 3, 3, 5, 5, 13, 13, 27, 21, 41, 41, 77, 63, 143, 129, 241, 203, 385, 347, 617, 491, 947, 835, 1445, 1185, 2511, 1991, 3585, 2915, 5411, 4569, 8063, 6321, 11131, 10133, 16465, 13207, 23817, 20133, 33929, 26663, 48357, 41363, 69605, 54363, 95727, 81183, 132257, 106581

Offset: 0

Gus Wiseman, May 13 2019

nonn

A composition of n is a finite sequence of positive integers summing to n.
A circular subinterval is a sequence of consecutive indices where the first and last indices are also considered consecutive.
For n > 0, a(n) is the number of subsets of Z_n which contain 0 and such that every ordered pair of distinct elements has a different difference (modulo n). The elements of a subset correspond with the partial sums of a composition. For example, when n = 8 the subset {0,2,7} corresponds with the composition (251). - Andrew Howroyd, Mar 24 2025

			The a(1) = 1 through a(8) = 13 compositions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)
            (12)  (13)  (14)  (15)  (16)   (17)
            (21)  (31)  (23)  (24)  (25)   (26)
                        (32)  (42)  (34)   (35)
                        (41)  (51)  (43)   (53)
                                    (52)   (62)
                                    (61)   (71)
                                    (124)  (125)
                                    (142)  (152)
                                    (214)  (215)
                                    (241)  (251)
                                    (412)  (512)
                                    (421)  (521)

Andrew Howroyd, Table of n, a(n) for n = 0..80

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

Cf. A325545, A325676, A325677, A325678, A325680, A325681, A325687, A382399.

suball[q_]:=Join[Take[q,#]&/@Select[Tuples[Range[Length[q]],2],OrderedQ],Drop[q,#]&/@Select[Tuples[Range[2,Length[q]-1],2],OrderedQ]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Total/@suball[#]&]],{n,0,15}]

a(n)={
   my(recurse(k,b,w)=
      if(k >= n, 1,
         b+=1<Andrew Howroyd, Mar 24 2025

a(21) onwards from Andrew Howroyd, Mar 24 2025

A382400 Number of subsets of Z_n such that every ordered pair of distinct elements has a different sum.

Original entry on oeis.org

1, 2, 4, 8, 15, 26, 48, 78, 133, 202, 316, 474, 755, 1054, 1604, 2196, 3305, 4370, 6208, 8228, 11631, 15086, 20912, 26842, 37581, 46626, 64052, 79984, 109635, 133314, 176156, 217094, 291409, 343872, 457828, 547576, 718375, 852074, 1112128, 1308230, 1714741

Offset: 0

Andrew Howroyd, Mar 27 2025

nonn

Arithmetic is done modulo n.

Every subset of size at most 3 is included. The cake numbers A000125 give the number of such subsets.

			The a(6) = 48 subsets are 42 subsets of size at most 3 and the following 6: {1,3,4,5}, {1,2,3,5}, {0,2,4,5}, {0,2,3,4}, {0,1,3,5}, {0,1,2,4}. Each of the size 4 subsets is perfect in the sense that every number from 0..5 can be written as the sum of two elements modulo 6 in exactly one way.

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A000125, A196723, A382399.

a(n)={
   my(recurse(k,r,b,w)=
      if(k >= n, 1,
         my(t=bitand((1<

cp's OEIS Frontend

A325679 Number of compositions of n such that every restriction to a circular subinterval has a different sum.

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