A325676 Number of compositions of n such that every distinct consecutive subsequence has a different sum.

1, 1, 2, 4, 5, 10, 12, 24, 26, 47, 50, 96, 104, 172, 188, 322, 335, 552, 590, 938, 1002, 1612, 1648, 2586, 2862, 4131, 4418, 6718, 7122, 10332, 11166, 15930, 17446, 24834, 26166, 37146, 41087, 55732, 59592, 84068, 89740, 122106, 133070, 177876, 194024, 262840, 278626

Offset: 0

Author

Gus Wiseman, May 13 2019

Keywords

• nonn

Comments

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

Compare to the definition of knapsack partitions (A108917).

Examples

			The distinct consecutive subsequences of (1,4,4,3) together with their sums are:
   1: {1}
   3: {3}
   4: {4}
   5: {1,4}
   7: {4,3}
   8: {4,4}
   9: {1,4,4}
  11: {4,4,3}
  12: {1,4,4,3}
Because the sums are all different, (1,4,4,3) is counted under a(12).
The a(1) = 1 through a(6) = 12 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (1111)  (41)     (42)
                            (113)    (51)
                            (122)    (114)
                            (221)    (132)
                            (311)    (222)
                            (11111)  (231)
                                     (411)
                                     (111111)
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A000079, A103295, A108917, A169942, A235998, A321143.

Cf. A325466, A325545, A325680, A325682, A325685, A325687, A325688.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Total/@Union[ReplaceList[#,{_,s__,_}:>{s}]]&]],{n,0,15}]

Extensions

a(21)-a(22) from Jinyuan Wang, Jun 20 2020

a(23)-a(25) from Robert Price, Jun 19 2021

a(26)-a(46) from Fausto A. C. Cariboni, Feb 10 2022