A295292 Sum of the products of the smaller and larger parts of the partitions of n into two parts with the larger part odd.

0, 1, 0, 3, 6, 14, 10, 22, 34, 55, 46, 73, 100, 140, 124, 172, 220, 285, 260, 335, 410, 506, 470, 578, 686, 819, 770, 917, 1064, 1240, 1176, 1368, 1560, 1785, 1704, 1947, 2190, 2470, 2370, 2670, 2970, 3311, 3190, 3553, 3916, 4324, 4180, 4612, 5044, 5525

Offset

1,4

Comments

Sum of the areas of the distinct rectangles with odd length and integer width such that L + W = n, W <= L. For example, a(8) = 22; the rectangles are 1 X 7 and 3 X 5, so 1*7 + 3*5 = 22.

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

Links

Table of n, a(n) for n=1..50.

Index entries for sequences related to partitions

Formula

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

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

G.f.: x^2*(1 - x + 3*x^2 + 3*x^3 + 5*x^4 - x^5 + 3*x^6 + 3*x^7) / ((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)*((2-2*(-1)^n)*(1+(-1)^n+6*(-1)^((2*n-3-(-1)^n)/4))+16*n*(-1+3*(-1)^n)+12*n^2*(1+(-1)^n+2*(-1)^((1/4)*(3+(-1)^n))*(-1)^(3*n/2))+16*n^3). - Wesley Ivan Hurt, Dec 03 2017

Example

a(8) = 22; the partitions of 8 into 2 parts are (7,1), (6,2), (5,3) and (4,4). Two of these partitions have odd numbers as their larger part: (7,1) and (5,3), so 7*1 + 5*3 = 22.

Maple

A295292:=n->add(i*(n-i)*((n-i) mod 2), i=1..floor(n/2)): seq(A295292(n), n=1..100);

Mathematica

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

Crossrefs

Cf. A295293.

Sequence in context: A301656 A359669 A322183 * A053365 A101034 A342103

Adjacent sequences: A295289 A295290 A295291 * A295293 A295294 A295295

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Nov 19 2017

Status

approved