A295318 Sum of the products of the smaller and larger parts of the partitions of n into two distinct parts with the smaller part even.

0, 0, 0, 0, 6, 8, 10, 12, 34, 40, 46, 52, 100, 112, 124, 136, 220, 240, 260, 280, 410, 440, 470, 500, 686, 728, 770, 812, 1064, 1120, 1176, 1232, 1560, 1632, 1704, 1776, 2190, 2280, 2370, 2460, 2970, 3080, 3190, 3300, 3916, 4048, 4180, 4312, 5044, 5200, 5356

Offset

1,5

Comments

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.

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).

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

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Index entries for sequences related to partitions

Formula

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

Conjectures from Colin Barker, Nov 20 2017: (Start)

G.f.: 2*x^5*(3 + x + x^2 + x^3 + 2*x^4) / ((1 - x)^4*(1 + x)^3*(1 + x^2)^3).

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.

(End)

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

Example

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

Maple

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

Mathematica

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

Prog

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

Crossrefs

Cf. A295317.

Sequence in context: A315852 A155776 A243826 * A343569 A331549 A184111

Adjacent sequences: A295315 A295316 A295317 * A295319 A295320 A295321

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Nov 19 2017

Status

approved