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A325688
Number of length-3 compositions of n such that every distinct consecutive subsequence has a different sum.
8
0, 0, 0, 1, 0, 4, 5, 12, 12, 25, 24, 40, 41, 60, 60, 85, 84, 112, 113, 144, 144, 181, 180, 220, 221, 264, 264, 313, 312, 364, 365, 420, 420, 481, 480, 544, 545, 612, 612, 685, 684, 760, 761, 840, 840, 925, 924, 1012, 1013, 1104, 1104, 1201, 1200, 1300, 1301, 1404
OFFSET
0,6
COMMENTS
A composition of n is a finite sequence of positive integers summing to n.
Confirmed recurrence relation from Colin Barker for n <= 5000. - Fausto A. C. Cariboni, Feb 13 2022
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 0..5000
FORMULA
Conjectures from Colin Barker, May 16 2019: (Start)
G.f.: x^3*(1 + 2*x^2 + 4*x^3 + 5*x^4) / ((1 - x)^3*(1 + x)^2*(1 + x + x^2)).
a(n) = 2*a(n-2) + a(n-3) - a(n-4) - 2*a(n-5) + a(n-7) for n>7.
(End)
EXAMPLE
The a(3) = 1 through a(8) = 12 compositions:
(111) (113) (114) (115) (116)
(122) (132) (124) (125)
(221) (222) (133) (143)
(311) (231) (142) (152)
(411) (214) (215)
(223) (233)
(241) (251)
(322) (332)
(331) (341)
(412) (512)
(421) (521)
(511) (611)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n, {3}], UnsameQ@@Total/@Union[ReplaceList[#, {___, s__, ___}:>{s}]]&]], {n, 0, 30}]
CROSSREFS
Column k = 3 of A325687.
Cf. A000217 (all length-3).
Sequence in context: A034773 A323766 A330223 * A080277 A047608 A266725
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 15 2019
STATUS
approved