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

0,6

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

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

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

Sequence in context: A096038 A064462 A248266 * A199890 A198459 A083581

Adjacent sequences: A325686 A325687 A325688 * A325690 A325691 A325692

Keyword

nonn

Author

Gus Wiseman, May 15 2019

Status

approved