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A325552
Number of compositions of n with distinct differences up to sign.
6
1, 1, 2, 3, 6, 9, 12, 23, 38, 61, 78, 135, 194, 315, 454, 699, 982, 1495, 2102, 3085, 4406, 6583, 9048, 13117, 18540, 26399, 36484, 51885, 72498, 100031, 139342, 192621, 267068, 367631, 505954, 687153, 946412, 1283367, 1745974, 2356935, 3207554, 4311591, 5816404
OFFSET
0,3
COMMENTS
A composition of n is a finite sequence of positive integers summing to n.
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).
a(n) has the same parity as n for n > 0, since reversing a composition does not change whether or not it has this property, and the only valid symmetric compositions are (n) and (n/2,n/2), with the latter only existing for even n. - Charlie Neder, Jun 06 2019
EXAMPLE
The differences of (1,2,1) are (1,-1), which are different but not up to sign, so (1,2,1) is not counted under a(4).
The a(1) = 1 through a(7) = 23 compositions:
(1) (2) (3) (4) (5) (6) (7)
(11) (12) (13) (14) (15) (16)
(21) (22) (23) (24) (25)
(31) (32) (33) (34)
(112) (41) (42) (43)
(211) (113) (51) (52)
(122) (114) (61)
(221) (132) (115)
(311) (213) (124)
(231) (133)
(312) (142)
(411) (214)
(223)
(241)
(322)
(331)
(412)
(421)
(511)
(1132)
(2113)
(2311)
(3112)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], UnsameQ@@Abs[Differences[#]]&]], {n, 0, 15}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 11 2019
EXTENSIONS
a(26)-a(42) from Alois P. Heinz, Jan 27 2024
STATUS
approved