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%I #18 Jan 21 2019 12:34:08
%S 0,1,2,3,4,8,10,12,14,21,24,27,30,40,44,48,52,65,70,75,80,96,102,108,
%T 114,133,140,147,154,176,184,192,200,225,234,243,252,280,290,300,310,
%U 341,352,363,374,408,420,432,444,481,494,507,520,560,574,588,602
%N Sum of the larger parts of the partitions of n into two parts such that the smaller part is odd.
%C Sum of the lengths of the distinct rectangles with integer length and odd width such that L + W = n, W <= L. For example, a(6) = 8; the rectangles with odd width are 1 X 5 and 3 X 3, and the sum of their lengths gives 5 + 3 = 8. - _Wesley Ivan Hurt_, Dec 06 2017
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{i=1..floor(n/2)} (n-i) * (i mod 2).
%F Conjectures from _Colin Barker_, Dec 06 2017: (Start)
%F G.f.: x^2*(1 + x + x^2 + x^3 + 2*x^4) / ((1 - x)^3*(1 + x)^2*(1 + x^2)^2).
%F a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9) for n>9.
%F (End)
%F a(n) = (1/64)*(12*n^2+4*n-6-4*(-1)^((2*n-3-(-1)^n)/4)+(4*n-2)*(-1)^n+(8*n-4)*(-1)^((2*n+3+(-1)^n)/4)). - _Wesley Ivan Hurt_, Dec 06 2017
%e a(10) = 21; the partitions of 10 into two parts are 9 + 1, 8 + 2, 7 + 3, 6 + 4 and 5 + 5. Of these, three have an odd smaller part and the sum of the larger parts of these partitions gives 9 + 7 + 5 = 21.
%t Table[Sum[(n - i) Mod[i, 2], {i, Floor[n/2]}], {n, 80}]
%t Table[Total[Select[IntegerPartitions[n,{2}],OddQ[#[[2]]]&][[All,1]]],{n,60}] (* _Harvey P. Dale_, Jan 21 2019 *)
%Y Cf. A295286, A295287.
%K nonn,easy
%O 1,3
%A _Wesley Ivan Hurt_, Dec 06 2017
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