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A295287 Sum of the products of the smaller and larger parts of the partitions of n into two parts with the smaller part even. 7

%I

%S 0,0,0,4,6,8,10,28,34,40,46,88,100,112,124,200,220,240,260,380,410,

%T 440,470,644,686,728,770,1008,1064,1120,1176,1488,1560,1632,1704,2100,

%U 2190,2280,2370,2860,2970,3080,3190,3784,3916,4048,4180,4888,5044,5200

%N Sum of the products of the smaller and larger parts of the partitions of n into two parts with the smaller part even.

%C Sum of the areas of the distinct rectangles with integer length and even width such that L + W = n, W <= L. For example, a(8) = 28; the rectangles are 2 X 6 and 4 X 4, so 2*6 + 4*4 = 28.

%C Sum of the ordinates from the ordered pairs (k,n*k-k^2) corresponding to integer points along the left side of the parabola b_k = n*k-k^2 where k is an even integer such that 0 < k <= floor(n/2).

%H Robert Israel, <a href="/A295287/b295287.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F a(n) = Sum_{i=1..floor(n/2)} i * (n - i) * ((i+1) mod 2).

%F Conjectures from _Colin Barker_, Nov 20 2017: (Start)

%F G.f.: 2*x^4*(2 + x + x^2 + x^3 + 3*x^4) / ((1 - x)^4*(1 + x)^3*(1 + x^2)^3).

%F a(n) = a(n-1) + 3*a(n-4) - 3*a(n-5) - 3*a(n-8) + 3*a(n-9) + a(n-12) - a(n-13) for n > 13.

%F (End)

%F Conjectures verified by _Robert Israel_, Dec 05 2017.

%F a(n) = (1/96)*(3*n^2*(1+(-1)^n)-16*n+4*n^3+3*I^n*(1+I)*(-1)^(-(-1)^n/4)*((1+2*n^2)*(-1)^n-1)/sqrt(2)) where I=sqrt(-1). - _Wesley Ivan Hurt_, Dec 02 2017

%e a(10) = 40; the partitions of 10 into two parts are (9,1), (8,2), (7,3), (6,4), (5,5). Two of these partitions have their smaller part even, namely (8,2) and (6,4). So, a(10) = 8*2 + 6*4 = 40.

%p A295287:=n->add(i*(n-i)*((i+1) mod 2), i=1..floor(n/2)): seq(A295287(n), n=1..100);

%p # Alternative:

%p for j from 0 to 3 do

%p F[j]:= expand(simplify(eval(sum(2*i*(4*k+j-2*i),i=1..k)),{k=(n-j)/4}))

%p od:

%p seq(F[n mod 4],n=1..30); # _Robert Israel_, Dec 05 2017

%t Table[Sum[i (n - i) Mod[i + 1, 2], {i, Floor[n/2]}], {n, 80}]

%Y Cf. A295286.

%K nonn,easy

%O 1,4

%A _Wesley Ivan Hurt_, Nov 19 2017

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Last modified May 21 15:32 EDT 2019. Contains 323444 sequences. (Running on oeis4.)