A308026 a(n) = n*(2*n - 3 - (-1)^n)*(11*n + (-1)^n)/24.

0, 0, 16, 30, 90, 134, 266, 356, 588, 740, 1100, 1330, 1846, 2170, 2870, 3304, 4216, 4776, 5928, 6630, 8050, 8910, 10626, 11660, 13700, 14924, 17316, 18746, 21518, 23170, 26350, 28240, 31856, 34000, 38080, 40494, 45066, 47766, 52858, 55860, 61500, 64820

Offset

1,3

Comments

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).

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).

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.

Links

Table of n, a(n) for n=1..42.

Index entries for sequences related to partitions

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

Formula

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

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).

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

Mathematica

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

Crossrefs

Sequence in context: A064634 A255265 A183372 * A145581 A186453 A129617

Adjacent sequences: A308023 A308024 A308025 * A308027 A308028 A308029

Keyword

nonn,easy

Author

Wesley Ivan Hurt, May 09 2019

Status

approved