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

0,6

Comments

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

Links

Table of n, a(n) for n=0..63.

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

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

Example

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 ).

Mathematica

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

Crossrefs

Cf. A036410, A069905.

Sequence in context: A089676 A334653 A230059 * A302486 A266745 A373219

Adjacent sequences: A340442 A340443 A340444 * A340446 A340447 A340448

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jan 07 2021

Status

approved