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 15 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 - x + 4*x^2) / ((1 - x)^3*(1 + x)^2) for n>5.
a(n) = -(5 + 3*(-1)^n - 2*n) * (n-2) / 4 for n>0.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
(End)
EXAMPLE
The a(3) = 1 through a(8) = 12 compositions (empty columns not shown):
(111) (113) (114) (115) (116)
(122) (141) (124) (125)
(131) (222) (133) (152)
(212) (411) (142) (161)
(221) (151) (215)
(311) (214) (233)
(223) (251)
(232) (323)
(241) (332)
(313) (512)
(322) (521)
(331) (611)
(412)
(421)
(511)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n, {3}], And@@Table[#[[i]]!=Total[Delete[#, i]], {i, 3}]&]], {n, 0, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 15 2019
STATUS
approved