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.

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

Links

Table of n, a(n) for n=1..43.

Sela Fried, On the ordinary generating function of A294139 and A307684, 2024.

Sela Fried, Proofs of some Conjectures from the OEIS, arXiv:2410.07237 [math.NT], 2024. See p. 10.

Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

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

Adjacent sequences: A294136 A294137 A294138 * A294140 A294141 A294142

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 31 2017

Extensions

Signature for linear recurrence taken from first formula in formula section.

Status

approved