A295317 Sum of the products of the smaller and larger parts of the partitions of n into two distinct parts with the smaller part odd.

0, 0, 2, 3, 4, 5, 18, 22, 26, 30, 64, 73, 82, 91, 156, 172, 188, 204, 310, 335, 360, 385, 542, 578, 614, 650, 868, 917, 966, 1015, 1304, 1368, 1432, 1496, 1866, 1947, 2028, 2109, 2570, 2670, 2770, 2870, 3432, 3553, 3674, 3795, 4468, 4612, 4756, 4900, 5694

Offset

1,3

Comments

Sum of the areas of the distinct rectangles with integer length and odd width such that L + W = n, W < L. For example, a(10) = 30; the rectangles are 1 X 9 and 3 X 7 (5 X 5 is not included since we have W < L), so 1*9 + 3*7 = 30.

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 odd 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 odd 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=8, i=1,3 so a(8) = 1*(8-1) + 3*(8-3) = 7 + 15 = 22. - Wesley Ivan Hurt, Mar 21 2018

Sum of the areas of the symmetric L-shaped polygons with long side n/2 and odd 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=7, i=1,3 so 1(7-1) + 3(7-3) = 6 + 12 = 18. - 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 mod 2).

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

G.f.: x^3*(2 + x + x^2 + x^3 + 7*x^4 + x^5 + x^6 + x^7 + x^8) / ((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)^(1/4)*I^n+(-1)^((2-(-1)^n)/4))+4*n*(6*n*(-1)^(1/4)*I^n+(-1)^((-1)^n/4)*(8-3*n*(1+(-1)^n)+4*n^2))) where I=sqrt(-1). - Wesley Ivan Hurt, Dec 02 2017

Example

For n=8, the partitions are 3 + 5 and 1 + 7, so a(8) = 3*5 + 1*7 = 22.

Maple

A295317:=n->add(i*(n-i)*(i mod 2), i=1..floor((n-1)/2)): seq(A295317(n), n=1..100);

Mathematica

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

Prog

(PARI) a(n) = sum(i=1, floor((n-1)/2), (i%2)*i*(n-i)) \\ Michael B. Porter, Dec 05 2017

Crossrefs

Cf. A295318.

Sequence in context: A145520 A337559 A061412 * A165646 A261639 A333993

Adjacent sequences: A295314 A295315 A295316 * A295318 A295319 A295320

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Nov 19 2017

Status

approved