A294139 - OEIS

%I #49 Oct 14 2024 11:21:50

%S 0,0,12,23,70,105,210,282,468,590,880,1065,1482,1743,2310,2660,3400,

%T 3852,4788,5355,6510,7205,8602,9438,11100,12090,14040,15197,17458,

%U 18795,21390,22920,25872,27608,30940,32895,36630,38817,42978,45410,50020,52710,57792

%N 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.

%H Sela Fried, <a href="/A294139/a294139.pdf">On the ordinary generating function of A294139 and A307684</a>, 2024.

%H Sela Fried, <a href="https://arxiv.org/abs/2410.07237">Proofs of some Conjectures from the OEIS</a>, arXiv:2410.07237 [math.NT], 2024. See p. 10.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).

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

%F Conjectures from _Colin Barker_, Nov 01 2017: (Start)

%F G.f.: x^3*(12 + 11*x + 11*x^2 + 2*x^3) / ((1 - x)^4*(1 + x)^3).

%F a(n) = n*(6*n - 1)*(n - 2) / 8 for n even.

%F a(n) = n*(3*n - 1)*(n - 1) / 4 for n odd.

%F 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)

%F a(n) = n*(4-21*n+12*n^2-5*n*(-1)^n)/16. - _Wesley Ivan Hurt_, Dec 02 2023

%F The first three conjectures of Barker are true. See links. - _Sela Fried_, Aug 11 2024.

%t Table[ Sum[2 i^2 + 2 (n - i)^2 + i (n - i), {i, Floor[(n-1)/2]}], {n, 40}]

%o (Magma) [n*(4-21*n+12*n^2-5*n*(-1)^n)/16 : n in [1..60]]; // _Wesley Ivan Hurt_, Dec 02 2023

%Y Cf. A294473.

%K nonn,easy

%O 1,3

%A _Wesley Ivan Hurt_, Oct 31 2017

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