A302647 a(n) = (2*n^2*(n^2 - 3) - (2*n^2 + 1)*(-1)^n + 1)/64.

0, 0, 2, 6, 18, 36, 72, 120, 200, 300, 450, 630, 882, 1176, 1568, 2016, 2592, 3240, 4050, 4950, 6050, 7260, 8712, 10296, 12168, 14196, 16562, 19110, 22050, 25200, 28800, 32640, 36992, 41616, 46818, 52326, 58482, 64980, 72200, 79800, 88200, 97020, 106722

Offset: 1

Author

Wesley Ivan Hurt, Apr 10 2018

Keywords

• nonn
• easy

Comments

Consider the partitions of n into two parts (s,t) where s <= t. Then a(n) represents the total volume of all rectangular prisms with dimensions s, t, and |t-s|.

Take a chessboard of (n+1) X (n+1) unit squares in which the a1 square is black. a(n) is the number of composite rectangles of p X q unit squares whose vertices are covered by white unit squares (1 < p <= n+1, 1 < q <= n+1). For example, in a 4 X 4 chessboard there are two such rectangles (for both rectangles p = q = 3) and the coordinates of their lower left vertices are a2 and b1, i.e., a(3) = 2. For the number of composite rectangles whose vertices are covered by black unit squares see A317714. - Ivan N. Ianakiev, Aug 22 2018

Also the graph crossing number of the (n+2)-barbell graph (assuming Guy's conjecture). - Eric W. Weisstein, May 17 2023

Links

• Eric Weisstein's World of Mathematics, Barbell Graph.
• Eric Weisstein's World of Mathematics, Graph Crossing Number.
• Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).
• Index entries for sequences related to partitions.

Crossrefs

Cf. A028723, A317714.

Positive terms are the third column of the triangle in A145118.

Programs

Magma

[(1/2)*Floor(n/2)*(1+Floor(n/2))*(Floor(n/2)-n)*(1-n+Floor(n/2)): n in [1..45]]; // Vincenzo Librandi, Apr 11 2018

Mathematica

Table[(1/2)*Floor[n/2]*(1 + Floor[n/2])*(Floor[n/2] - n)*(1 - n + Floor[n/2]), {n, 100}]
LinearRecurrence[{2, 2, -6, 0, 6, -2, -2, 1}, {0, 0, 2, 6, 18, 36, 72, 120}, 20] (* Eric W. Weisstein, May 17 2023 *)
Table[(1 - (-1)^n - 2 (3 + (-1)^n) n^2 + 2 n^4)/64, {n, 20}] (* Eric W. Weisstein, May 17 2023 *)
CoefficientList[Series[-2 x^2 (1 + x + x^2)/((-1 + x)^5 (1 + x)^3), {x, 0, 20}], x] (* Eric W. Weisstein, May 17 2023 *)

Formula

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

a(n) = (1/2)*floor(n/2)*(1+floor(n/2))*(floor(n/2)-n)*(1-n+floor(n/2)).

From Colin Barker, Apr 11 2018: (Start)

G.f.: 2*x^3*(1 + x + x^2) / ((1 - x)^5*(1 + x)^3).

a(n) = n^2*(n-2)*(n+2) / 32 for n even.

a(n) = (n^2 - 1)^2 / 32 for n odd.

a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8) for n>8. (End)

a(n) = 2 * A028723(n+2). - Alois P. Heinz, Apr 12 2018

a(n) = 2 * binomial(floor((n+1)/2),2) * binomial(floor((n+2)/2),2). - Bruno Berselli, Apr 12 2018

Sum_{n>=3} 1/a(n) = Pi^2/3 - 5/2. - Amiram Eldar, Jun 20 2025