A303120 Total area of all rectangles of size p X q such that p + q = n^2 and p <= q.

0, 7, 60, 372, 1300, 4047, 9800, 22352, 44280, 84575, 147620, 251412, 402220, 632247, 949200, 1406272, 2011440, 2847447, 3920460, 5353300, 7147140, 9477567, 12336280, 15966672, 20345000, 25800047, 32284980, 40234292, 49568540, 60851175, 73958560, 89609472

Offset: 1

Author

Wesley Ivan Hurt, Apr 18 2018

Keywords

• nonn
• easy

Comments

Sum of all the products formed using the corresponding largest and smallest parts of each partition of n^2 into two parts. - Wesley Ivan Hurt, Mar 26 2019

Examples

			a(3) = 60; The rectangles are 8 X 1, 7 X 2, 6 X 3 and 5 X 4. The total area is then 8*1 + 7*2 + 6*3 + 5*4 = 60.
a(4) = 372; The rectangles are 15 X 1, 14 X 2, 13 X 3, 12 X 4, 11 X 5, 10 X 6, 9 X 7 and 8 X 8. The total area of the rectangles is then 15*1 + 14*2 + 13*3 + 12*4 + 11*5 + 10*6 + 9*7 + 8*8 = 372.
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..2000
• Index entries for sequences related to partitions

Crossrefs

Cf. A000290, A023855.

Programs

GAP

List([1..35],n->Sum([1..Int(n^2/2)],i->i*(n^2-i))); # Muniru A Asiru, Mar 15 2019

Magma

[&+[i*(n^2-i): i in [0..Floor(n^2/2)]]: n in [1..35]]; // Vincenzo Librandi, Apr 19 2018

Maple

A303120:=n->add(i*(n^2-i), i=1..floor(n^2/2)): seq(A303120(n), n=1..50); # Wesley Ivan Hurt, Mar 12 2019

Mathematica

Table[Sum[i*(n^2 - i), {i, Floor[n^2/2]}], {n, 50}]

PARI

a(n) = sum(i=1, n^2\2, i*(n^2-i)); \\ Michel Marcus, Mar 13 2019

Formula

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

Conjectures from Colin Barker, Apr 19 2018 and Mar 19 2019: (Start)

G.f.: x^2*(7 + 46*x + 224*x^2 + 386*x^3 + 594*x^4 + 386*x^5 + 224*x^6 + 46*x^7 + 7*x^8) / ((1 - x)^7*(1 + x)^5).

a(n) = 2*a(n-1) + 4*a(n-2) - 10*a(n-3) - 5*a(n-4) + 20*a(n-5) - 20*a(n-7) + 5*a(n-8) + 10*a(n-9) - 4*a(n-10) - 2*a(n-11) + a(n-12) for n > 12.

a(n) = (n^2*(-4 + 3*(1+(-1)^n)*n^2 + 4*n^4)) / 48.

(End)