A340445 - OEIS

%I #29 Dec 22 2024 16:22:02

%S 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,

%T 56,60,65,70,74,80,85,90,96,102,107,114,120,126,133,140,146,154,161,

%U 168,176,184,191,200,208,216,225,234,242,252,261,270,280,290,299,310,320,330

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

%C Conjecturally the same as A230059 (apart from the offset). - _R. J. Mathar_, Jan 14 2021

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).

%F 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.

%F 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.

%F From _Alois P. Heinz_, Jan 07 2021: (Start)

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

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

%F a(n) = A036410(n-1)-1. - _Hugo Pfoertner_, Jan 09 2021

%F a(n) + A079978(n) = A069905(n), n>0. - _R. J. Mathar_, Jan 18 2021

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

%e a(6) = 2; [4,1,1], [3,2,1] ( [2,2,2] not counted ),

%e a(7) = 4; [5,1,1], [4,2,1], [3,3,1], [3,2,2],

%e a(8) = 5; [6,1,1], [5,2,1], [4,3,1], [4,2,2], [3,3,2],

%e 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 ).

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

%Y Cf. A036410, A069905.

%K nonn,easy

%O 0,6

%A _Wesley Ivan Hurt_, Jan 07 2021