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
Links
- 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).
Crossrefs
Cf. A294473.
Programs
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Magma
[n*(4-21*n+12*n^2-5*n*(-1)^n)/16 : n in [1..60]]; // Wesley Ivan Hurt, Dec 02 2023
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Mathematica
Table[ Sum[2 i^2 + 2 (n - i)^2 + i (n - i), {i, Floor[(n-1)/2]}], {n, 40}]
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.
Extensions
Signature for linear recurrence taken from first formula in formula section.