A325686 Number of strict length-3 compositions x + y + z = n satisfying x + y != z and x != y + z.

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

Author

Gus Wiseman, May 13 2019

Keywords

• nonn

Comments

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

From Kevin O'Bryant, Jun 02 2025: (Start)

Also the number of Sidon sets in {0,1,...,n} with 4 elements that contain both 0 and n.

Also, the number of 3-tuples of positive integers with the 6 numbers x, y, z, x+y, y+z, x+y+z=n all distinct. (End)

Examples

			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)
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..5000

Crossrefs

Column k = 3 of A325677.

Cf. A000079, A001399, A005044, A108917, A124278, A143823, A266223, A275972.

Cf. A325676, A325688, A325689, A325690, A325691.

Programs

Mathematica

Table[Length[Cases[Join@@Permutations/@IntegerPartitions[n,{3}],{x_,y_,z_}/;x!=y!=z&&x+y!=z &&x!=y+z]],{n,0,30}]

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