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%I #10 Apr 08 2018 09:21:57
%S 0,5,8,18,24,41,50,72,84,113,128,162,180,221,242,288,312,365,392,450,
%T 480,545,578,648,684,761,800,882,924,1013,1058,1152,1200,1301,1352,
%U 1458,1512,1625,1682,1800,1860,1985,2048,2178,2244,2381,2450,2592,2664,2813
%N Number of even parts in the partitions of 3n into 3 parts.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F Conjectures from _Colin Barker_, Apr 07 2018: (Start)
%F G.f.: x^2*(5 + 3*x + 5*x^2 + 3*x^3 + 2*x^4) / ((1 - x)^3*(1 + x)^2*(1 + x^2)).
%F a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7) for n>7.
%F (End)
%e Count the even parts for a(n) (n > 0).
%e 13 + 1 + 1
%e 12 + 2 + 1
%e 11 + 3 + 1
%e 10 + 4 + 1
%e 9 + 5 + 1
%e 8 + 6 + 1
%e 7 + 7 + 1
%e 10 + 1 + 1 11 + 2 + 2
%e 9 + 2 + 1 10 + 3 + 2
%e 8 + 3 + 1 9 + 4 + 2
%e 7 + 4 + 1 8 + 5 + 2
%e 6 + 5 + 1 7 + 6 + 2
%e 7 + 1 + 1 8 + 2 + 2 9 + 3 + 3
%e 6 + 2 + 1 7 + 3 + 2 8 + 4 + 3
%e 5 + 3 + 1 6 + 4 + 2 7 + 5 + 3
%e 4 + 4 + 1 5 + 5 + 2 6 + 6 + 3
%e 4 + 1 + 1 5 + 2 + 2 6 + 3 + 3 7 + 4 + 4
%e 3 + 2 + 1 4 + 3 + 2 5 + 4 + 3 6 + 5 + 4
%e 1 + 1 + 1 2 + 2 + 2 3 + 3 + 3 4 + 4 + 4 5 + 5 + 5
%e 3(1) 3(2) 3(3) 3(4) 3(5) .. 3n
%e ---------------------------------------------------------------------
%e 0 5 8 18 24 .. a(n)
%t Table[Count[Flatten[IntegerPartitions[3 n, {3}]], _?EvenQ], {n, 100}]
%Y Cf. A302392 (odd parts).
%K nonn
%O 1,2
%A _Wesley Ivan Hurt_, Apr 07 2018