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A302647 a(n) = (2*n^2*(n^2 - 3) - (2*n^2 + 1)*(-1)^n + 1)/64. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

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

LINKS

Table of n, a(n) for n=1..43.

Index entries for sequences related to partitions

FORMULA

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

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

MATHEMATICA

Table[(1/2)*Floor[n/2]*(1 + Floor[n/2])*(Floor[n/2] - n)*(1 - n + Floor[n/2]), {n, 100}]

PROG

(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

CROSSREFS

Cf. A028723.

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

Sequence in context: A034881 A146345 A064842 * A324580 A101695 A014741

Adjacent sequences:  A302644 A302645 A302646 * A302648 A302649 A302650

KEYWORD

nonn,easy

AUTHOR

Wesley Ivan Hurt, Apr 10 2018

STATUS

approved

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Last modified August 18 13:25 EDT 2019. Contains 326100 sequences. (Running on oeis4.)