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

0, 0, 0, 4, 6, 8, 10, 28, 34, 40, 46, 88, 100, 112, 124, 200, 220, 240, 260, 380, 410, 440, 470, 644, 686, 728, 770, 1008, 1064, 1120, 1176, 1488, 1560, 1632, 1704, 2100, 2190, 2280, 2370, 2860, 2970, 3080, 3190, 3784, 3916, 4048, 4180, 4888, 5044, 5200

Offset

1,4

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(8) = 28; the rectangles are 2 X 6 and 4 X 4, so 2*6 + 4*4 = 28.

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).

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Index entries for sequences related to partitions

Formula

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

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

G.f.: 2*x^4*(2 + x + x^2 + x^3 + 3*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)

Conjectures verified by Robert Israel, Dec 05 2017.

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

Example

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.

Maple

A295287:=n->add(i*(n-i)*((i+1) mod 2), i=1..floor(n/2)): seq(A295287(n), n=1..100);
# Alternative:
for j from 0 to 3 do
  F[j]:= expand(simplify(eval(sum(2*i*(4*k+j-2*i), i=1..k)), {k=(n-j)/4}))
od:
seq(F[n mod 4], n=1..30); # Robert Israel, Dec 05 2017

Mathematica

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

Crossrefs

Cf. A295286.

Sequence in context: A298473 A374787 A054284 * A351758 A079254 A091178

Adjacent sequences: A295284 A295285 A295286 * A295288 A295289 A295290

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Nov 19 2017

Status

approved