login
A325686
Number of strict length-3 compositions x + y + z = n satisfying x + y != z and x != y + z.
7
0, 0, 0, 0, 0, 0, 2, 6, 8, 18, 16, 30, 34, 48, 48, 72, 72, 96, 98, 126, 128, 162, 160, 198, 202, 240, 240, 288, 288, 336, 338, 390, 392, 450, 448, 510, 514, 576, 576, 648, 648, 720, 722, 798, 800, 882, 880, 966, 970, 1056, 1056, 1152, 1152, 1248, 1250, 1350, 1352
OFFSET
0,7
COMMENTS
A composition of n is a finite sequence of positive integers summing to n.
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 0..5000
FORMULA
Conjectures from Colin Barker, May 14 2019: (Start)
G.f.: 2*x^6*(1 + 3*x + 3*x^2 + 5*x^3) / ((1 - x)^3*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n>9.
(End)
Above conjecture confirmed for n <= 5000. - Fausto A. C. Cariboni, Feb 17 2022
EXAMPLE
The a(6) = 2 through a(10) = 16 compositions:
(132) (124) (125) (126) (127)
(231) (142) (143) (135) (136)
(214) (152) (153) (154)
(241) (215) (162) (163)
(412) (251) (216) (172)
(421) (341) (234) (217)
(512) (243) (253)
(521) (261) (271)
(315) (316)
(324) (352)
(342) (361)
(351) (451)
(423) (613)
(432) (631)
(513) (712)
(531) (721)
(612)
(621)
MATHEMATICA
Table[Length[Cases[Join@@Permutations/@IntegerPartitions[n, {3}], {x_, y_, z_}/; x!=y!=z&&x+y!=z &&x!=y+z]], {n, 0, 30}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 13 2019
STATUS
approved