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A332310 Number of compositions (ordered partitions) of n into distinct parts where no part is a multiple of 4.

%I A332310 #11 Nov 17 2020 05:38:13
%S A332310 1,1,1,3,2,3,9,5,12,17,23,43,50,55,67,111,144,273,291,377,410,689,827,
%T A332310 961,1860,1663,2647,3573,4610,4683,6753,8465,11232,16835,19985,24073,
%U A332310 29258,40411,51367,58431,72084,99807,119409,176387,199922,250841,290123
%N A332310 Number of compositions (ordered partitions) of n into distinct parts where no part is a multiple of 4.
%H A332310 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%e A332310 a(7) = 5 because we have [7], [6, 1], [5, 2], [2, 5] and [1, 6].
%p A332310 b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0,
%p A332310       p!, add(b(n-i*j, i-1, p+j), j=0..min(irem(i, 4), 1, n/i))))
%p A332310     end:
%p A332310 a:= n-> b(n$2, 0):
%p A332310 seq(a(n), n=0..55);  # _Alois P. Heinz_, Feb 09 2020
%t A332310 b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0, If[n == 0, p!, Sum[b[n - i j, i - 1, p + j], {j, 0, Min[Mod[i, 4], 1, n/i]}]]];
%t A332310 a[n_] := b[n, n, 0];
%t A332310 a /@ Range[0, 55] (* _Jean-François Alcover_, Nov 17 2020, after _Alois P. Heinz_ *)
%Y A332310 Cf. A001935, A032020, A032021, A070048, A332309, A332311.
%K A332310 nonn
%O A332310 0,4
%A A332310 _Ilya Gutkovskiy_, Feb 09 2020