A308026 - OEIS

%I #10 May 10 2019 11:07:36

%S 0,0,16,30,90,134,266,356,588,740,1100,1330,1846,2170,2870,3304,4216,

%T 4776,5928,6630,8050,8910,10626,11660,13700,14924,17316,18746,21518,

%U 23170,26350,28240,31856,34000,38080,40494,45066,47766,52858,55860,61500,64820

%N a(n) = n*(2*n - 3 - (-1)^n)*(11*n + (-1)^n)/24.

%C Total surface area of all rectangular prisms with dimensions s X t X t where s and t are positive integers, n = s + t and s < t. For example, the surface area gives 4*s*t + 2*t^2 = 2*t*(2*s+t).

%C Consider the partitions of n into two distinct parts (s,t) with s < t. Then a(n) is the sum of all the products (2*t)*(2*s+t), using corresponding parts from each (s,t).

%C Also, the total area of all rectangles with dimensions (2*t) X (2*s+t), where s and t are positive integers, n = s + t and s < t.

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%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 G.f.: 2*x^3*(8 + 7*x + 6*x^2 + x^3)/((1 + x)^3*(1 - x)^4). - _Bruno Berselli_, May 10 2019

%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).

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

%t Table[n*(2*n - 3 - (-1)^n)*(11*n + (-1)^n)/24, {n, 60}]

%K nonn,easy

%O 1,3

%A _Wesley Ivan Hurt_, May 09 2019