login
A294139
Sum of the areas of the distinct rectangles (and the areas of the squares on their sides) with positive integer sides such that L + W = n, W < L.
1
0, 0, 12, 23, 70, 105, 210, 282, 468, 590, 880, 1065, 1482, 1743, 2310, 2660, 3400, 3852, 4788, 5355, 6510, 7205, 8602, 9438, 11100, 12090, 14040, 15197, 17458, 18795, 21390, 22920, 25872, 27608, 30940, 32895, 36630, 38817, 42978, 45410, 50020, 52710, 57792
OFFSET
1,3
FORMULA
a(n) = Sum_{i=1..floor((n-1)/2)} 2*i^2 + 2*(n-i)^2 + i*(n-i).
Conjectures from Colin Barker, Nov 01 2017: (Start)
G.f.: x^3*(12 + 11*x + 11*x^2 + 2*x^3) / ((1 - x)^4*(1 + x)^3).
a(n) = n*(6*n - 1)*(n - 2) / 8 for n even.
a(n) = n*(3*n - 1)*(n - 1) / 4 for n odd.
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n > 7. (End)
a(n) = n*(4-21*n+12*n^2-5*n*(-1)^n)/16. - Wesley Ivan Hurt, Dec 02 2023
The first three conjectures of Barker are true. See links. - Sela Fried, Aug 11 2024.
MATHEMATICA
Table[ Sum[2 i^2 + 2 (n - i)^2 + i (n - i), {i, Floor[(n-1)/2]}], {n, 40}]
PROG
(Magma) [n*(4-21*n+12*n^2-5*n*(-1)^n)/16 : n in [1..60]]; // Wesley Ivan Hurt, Dec 02 2023
CROSSREFS
Cf. A294473.
Sequence in context: A190426 A207539 A083683 * A255766 A333933 A015447
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Oct 31 2017
EXTENSIONS
Signature for linear recurrence taken from first formula in formula section.
STATUS
approved