A340445 Number of partitions of n into 3 parts that are not all the same.

0, 0, 0, 0, 1, 2, 2, 4, 5, 6, 8, 10, 11, 14, 16, 18, 21, 24, 26, 30, 33, 36, 40, 44, 47, 52, 56, 60, 65, 70, 74, 80, 85, 90, 96, 102, 107, 114, 120, 126, 133, 140, 146, 154, 161, 168, 176, 184, 191, 200, 208, 216, 225, 234, 242, 252, 261, 270, 280, 290, 299, 310, 320, 330

Offset: 0

Author

Wesley Ivan Hurt, Jan 07 2021

Keywords

• nonn
• easy

Comments

Conjecturally the same as A230059 (apart from the offset). - R. J. Mathar, Jan 14 2021

Examples

			a(6) = 2; [4,1,1], [3,2,1] ( [2,2,2] not counted ),
a(7) = 4; [5,1,1], [4,2,1], [3,3,1], [3,2,2],
a(8) = 5; [6,1,1], [5,2,1], [4,3,1], [4,2,2], [3,3,2],
a(9) = 6; [7,1,1], [6,2,1], [5,3,1], [4,4,1], [5,2,2], [4,3,2] ( [3,3,3] not counted ).
		

Links

• Index entries for sequences related to partitions
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Crossrefs

Cf. A036410, A069905.

Programs

Mathematica

Table[Sum[Sum[(1 - KroneckerDelta[i, k, n - i - k]), {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 0, 80}]

Formula

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1 - [k = i = n-i-k]), where [ ] is the (generalized) Iverson bracket.

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1 - [k = i] * [2*i = n-k] * [2*k = n-i]), where [ ] is the Iverson bracket.

From Alois P. Heinz, Jan 07 2021: (Start)

G.f.: x^4*(x^2-x-1)/((x+1)*(x^2+x+1)*(x-1)^3).

a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6), n>6. (End)

a(n) = A036410(n-1)-1. - Hugo Pfoertner, Jan 09 2021

a(n) + A079978(n) = A069905(n), n>0. - R. J. Mathar, Jan 18 2021

72*a(n) = -16*A099837(n+3) -9*(-1)^n +6*n^2 -31. - R. J. Mathar, Jun 09 2022