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

%I A340445 #29 Dec 22 2024 16:22:02
%S A340445 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 A340445 56,60,65,70,74,80,85,90,96,102,107,114,120,126,133,140,146,154,161,
%U A340445 168,176,184,191,200,208,216,225,234,242,252,261,270,280,290,299,310,320,330
%N A340445 Number of partitions of n into 3 parts that are not all the same.
%C A340445 Conjecturally the same as A230059 (apart from the offset). - _R. J. Mathar_, Jan 14 2021
%H A340445 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H A340445 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).
%F A340445 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 A340445 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 A340445 From _Alois P. Heinz_, Jan 07 2021: (Start)
%F A340445 G.f.: x^4*(x^2-x-1)/((x+1)*(x^2+x+1)*(x-1)^3).
%F A340445 a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6), n>6. (End)
%F A340445 a(n) = A036410(n-1)-1. - _Hugo Pfoertner_, Jan 09 2021
%F A340445 a(n) + A079978(n) = A069905(n), n>0. - _R. J. Mathar_, Jan 18 2021
%F A340445 72*a(n) = -16*A099837(n+3) -9*(-1)^n +6*n^2 -31. - _R. J. Mathar_, Jun 09 2022
%e A340445 a(6) = 2; [4,1,1], [3,2,1] ( [2,2,2] not counted ),
%e A340445 a(7) = 4; [5,1,1], [4,2,1], [3,3,1], [3,2,2],
%e A340445 a(8) = 5; [6,1,1], [5,2,1], [4,3,1], [4,2,2], [3,3,2],
%e A340445 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 A340445 Table[Sum[Sum[(1 - KroneckerDelta[i, k, n - i - k]), {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 0, 80}]
%Y A340445 Cf. A036410, A069905.
%K A340445 nonn,easy
%O A340445 0,6
%A A340445 _Wesley Ivan Hurt_, Jan 07 2021