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%I #12 Sep 13 2020 14:45:36
%S 0,0,0,0,1,0,1,2,2,3,5,5,9,10,15,18,24,30,40,47,65,76,97,120,150,179,
%T 229,270,336,404,494,588,722,852,1035,1227,1476,1744,2090,2461,2935,
%U 3446,4092,4795,5665,6627,7801,9096,10680,12426,14528,16881,19680,22800,26525
%N Number of partitions of n with equal nonzero number of parts congruent to each of 0 and 1 (mod 3).
%p b := proc(n, i, t, s) option remember; `if`(n=0,
%p `if`(t = s and t >= 1, 1, 0), `if`(i<1, 0, b(n, i-1, t, s)+
%p b(n-i, min(n-i, i), t + [1, 0, 0][1+irem(i, 3)], s + [0, 1, 0][1+irem(i, 3)])))
%p end:
%p seq(b(n,n, 0,0), n=0..50); # _Georg Fischer_, Sep 13 2020
%t equalQ[partit_] := With[{t = Total[Switch[Mod[#, 3], 0, {1, 0, 0}, 1, {0, 1, 0}, 2, {0, 0, 1}]& /@ partit]}, t[[1]] > 0 && t[[1]] == t[[2]]]; a[n_] := Select[IntegerPartitions[n], equalQ] // Length; a[0] = 0; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 54}] (* _Jean-François Alcover_, Dec 07 2016 *)
%K nonn
%O 0,8
%A _Olivier Gérard_
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