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A286744 Number of distinct partitions of n with parts differing by at least two with smallest part at least two and with an even number of parts.

%I A286744 #24 Jun 06 2021 09:00:44
%S A286744 1,0,0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,7,9,9,11,12,15,16,20,22,27,30,
%T A286744 36,40,48,53,62,69,80,88,101,111,126,138,156,170,191,208,232,253,282,
%U A286744 306,340,370,410,446,494,537,594,647,714,778,859,935,1031,1124
%N A286744 Number of distinct partitions of n with parts differing by at least two with smallest part at least two and with an even number of parts.
%H A286744 Alois P. Heinz, <a href="/A286744/b286744.txt">Table of n, a(n) for n = 0..10000</a>
%F A286744 a(n) ~ exp(2*Pi*sqrt(n/15)) / (4 * 3^(1/4) * sqrt(5*phi) * n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Mar 10 2020
%e A286744 a(8) = 2 because 6+2 and 5+3 are the only partitions of 8 that satisfy the three conditions.
%p A286744 b:= proc(n, i, t) option remember; `if`(n=0, t,
%p A286744       `if`(i>n, 0, b(n, i+1, t)+b(n-i, i+2, 1-t)))
%p A286744     end:
%p A286744 a:= n-> b(n, 2, 1):
%p A286744 seq(a(n), n=0..80);  # _Alois P. Heinz_, Nov 23 2017
%t A286744 Table[Length@
%t A286744   Select[IntegerPartitions@n,
%t A286744    Min[-Differences@#] >= 2 && Min@# >= 2 && EvenQ@Length@# &], {n,
%t A286744   20}]
%t A286744 (* Second program: *)
%t A286744 b[n_, i_, t_] := b[n, i, t] = If[n == 0, t,
%t A286744      If[i > n, 0, b[n, i + 1, t] + b[n - i, i + 2, 1 - t]]];
%t A286744 a[n_] := b[n, 2, 1];
%t A286744 a /@ Range[0, 80] (* _Jean-François Alcover_, Jun 06 2021, after _Alois P. Heinz_ *)
%Y A286744 Cf. A286745, A003106, A224898, A001622.
%K A286744 nonn
%O A286744 0,9
%A A286744 _George Beck_, May 13 2017