A295293 Sum of the products of the smaller and larger parts of the partitions of n into two parts with the larger part even.

0, 0, 2, 4, 4, 8, 18, 28, 26, 40, 64, 88, 82, 112, 156, 200, 188, 240, 310, 380, 360, 440, 542, 644, 614, 728, 868, 1008, 966, 1120, 1304, 1488, 1432, 1632, 1866, 2100, 2028, 2280, 2570, 2860, 2770, 3080, 3432, 3784, 3674, 4048, 4468, 4888, 4756, 5200, 5694

Offset

1,3

Comments

Sum of the areas of the distinct rectangles with even length and integer 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 (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 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) * ((n-i+1) mod 2).

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

G.f.: 2*x^3*(1 + x + 2*x^3 + 2*x^4 + 2*x^5 - x^6 + 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)

Conjectures verified by Robert Israel, Dec 05 2017.

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)^((3+(-1)^n)/4)*(-1)^(3*n/2))+16*n^3). - Wesley Ivan Hurt, Dec 04 2017

a(n) = A295287(n) if n is even, A295286(n) if n is odd. - Robert Israel, Dec 05 2017

Example

a(8) = 28; the partitions of 8 into two parts are (7,1), (6,2), (5,3) and (4,4). There are two partitions with the larger part even: (6,2) and (4,4), so 6*2 + 4*4 = 28.

Maple

A295293:=n->add(i*(n-i)*((n-i+1) mod 2), i=1..floor(n/2)): seq(A295293(n), n=1..100);
# Alternative:
F[0]:= (1/24)*n^3+(1/8)*n^2-(1/6)*n:
F[1]:= -1/16+(1/24)*n^3-(1/16)*n^2+(1/12)*n:
F[2]:= (1/24)*n^3-(1/6)*n:
F[3]:= 1/16+(1/24)*n^3+(1/16)*n^2+(1/12)*n:
seq(F[n mod 4], n=1..100); # Robert Israel, Dec 05 2017

Mathematica

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

Crossrefs

Cf. A295286, A295287, A295292.

Sequence in context: A161816 A240347 A160124 * A080007 A239649 A264190

Adjacent sequences: A295290 A295291 A295292 * A295294 A295295 A295296

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Nov 19 2017

Status

approved