A325689 Number of length-3 compositions of n such that no part is the sum of the other two.

0, 0, 0, 1, 0, 6, 4, 15, 12, 28, 24, 45, 40, 66, 60, 91, 84, 120, 112, 153, 144, 190, 180, 231, 220, 276, 264, 325, 312, 378, 364, 435, 420, 496, 480, 561, 544, 630, 612, 703, 684, 780, 760, 861, 840, 946, 924, 1035, 1012, 1128, 1104, 1225, 1200, 1326, 1300, 1431

Offset: 0

Author

Gus Wiseman, May 15 2019

Keywords

• nonn

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

Examples

			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)
		

Links

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

Crossrefs

Cf. A000079, A001399, A005044, A008642, A069905, A124278, A266223.

Cf. A325676, A325688, A325690, A325691, A325694.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],And@@Table[#[[i]]!=Total[Delete[#,i]],{i,3}]&]],{n,0,30}]

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)