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A286745 Number of distinct partitions of n with parts differing by at least two with smallest part at least two and with an odd number of parts.

%I A286745 #25 Mar 10 2020 08:31:09
%S A286745 0,0,1,1,1,1,1,1,1,1,1,1,2,2,3,4,5,6,8,9,11,13,15,17,20,22,25,28,31,
%T A286745 34,39,42,47,52,58,64,72,79,89,99,111,123,139,154,173,193,216,240,269,
%U A286745 298,333,369,410,453,503,554,613,674,743,815,897,981,1077,1177,1288,1405,1536,1672,1825,1985,2163,2350,2558,2776,3019,3275,3557,3856,4186,4534,4919
%N A286745 Number of distinct partitions of n with parts differing by at least two with smallest part at least two and with an odd number of parts.
%H A286745 Alois P. Heinz, <a href="/A286745/b286745.txt">Table of n, a(n) for n = 0..10000</a>
%F A286745 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 A286745 a(12) = 2 because of the partitions of 12, 12 and 6+4+2 are the only two that satisfy all three conditions.
%p A286745 b:= proc(n, i, t) option remember; `if`(n=0, t,
%p A286745       `if`(i>n, 0, b(n, i+1, t)+b(n-i, i+2, 1-t)))
%p A286745     end:
%p A286745 a:= n-> b(n, 2, 0):
%p A286745 seq(a(n), n=0..80);  # _Alois P. Heinz_, Nov 23 2017
%t A286745 Table[Length@ Select[ip@n, Min[-Differences@#] >= 2 && Min@# >= 2 && OddQ@Length@# &], {n, 20}]
%Y A286745 Cf. A286744, A003106, A224898.
%K A286745 nonn
%O A286745 0,13
%A A286745 _George Beck_, May 13 2017