A295318 - OEIS

%I #44 Apr 16 2018 19:00:54

%S 0,0,0,0,6,8,10,12,34,40,46,52,100,112,124,136,220,240,260,280,410,

%T 440,470,500,686,728,770,812,1064,1120,1176,1232,1560,1632,1704,1776,

%U 2190,2280,2370,2460,2970,3080,3190,3300,3916,4048,4180,4312,5044,5200,5356

%N Sum of the products of the smaller and larger parts of the partitions of n into two distinct 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(12) = 52; the rectangles are 2 X 10 and 4 X 8 (6 X 6 is not included since we have W < L), so 2*10 + 4*8 = 52.

%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).

%C Sum of the areas of the trapezoids with bases n and n-2i and height i for even i in 0 <= i <= floor((n-1)/2). For a(n) the area formula for a trapezoid becomes (n+n-2i)*i/2 = (2n-2i)*i/2 = i*(n-i). For n=9, i=0,2,4 so a(9) = 0*(9-0) + 2*(9-2) + 4*(9-4) = 0 + 14 + 20 = 34. - _Wesley Ivan Hurt_, Mar 22 2018

%C Sum of the areas of the symmetric L-shaped polygons with long side n/2 and even width i in 0 <= i <= floor((n-1)/2). The area of each polygon is given by i^2+2i(n/2-i) = i^2+ni-2i^2 = i(n-i). For n=9, i=0,2,4 so 0(9-0) + 2(9-2) + 4(9-4) = 0 + 14 + 20 = 34. - _Wesley Ivan Hurt_, Mar 26 2018

%H Vincenzo Librandi, <a href="/A295318/b295318.txt">Table of n, a(n) for n = 1..5000</a>

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

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

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

%F G.f.: 2*x^5*(3 + x + x^2 + x^3 + 2*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 a(n) = (1/384)*(-1)^(-(-1)^n/4)*((-2+2*(-1)^n)*((-1)^((4*n+2-(-1)^n)/4)+6*(-1)^((2*n+1)/4)+(-1)^((2-(-1)^n)/4))+4*n*(-6*n*(-1)^((2*n+1)/4)+(-1)^((-1)^n/4)*(-16-3*n*(1+(-1)^n)+4*n^2))). - _Wesley Ivan Hurt_, Dec 02 2017

%e For n=9, the partitions are 7 + 2 and 5 + 4, so a(9) = 7*2 + 5*4 = 34. - _Michael B. Porter_, Dec 05 2017

%p A295318:=n->add(i*(n-i)*((i+1) mod 2), i=1..floor((n-1)/2)): seq(A295318(n), n=1..100);

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

%o (PARI) a(n) = sum(i=1, (n-1)\2, i*((i+1)%2)*(n-i)); \\ _Altug Alkan_, Mar 22 2018

%Y Cf. A295317.

%K nonn,easy

%O 1,5

%A _Wesley Ivan Hurt_, Nov 19 2017