A279034 The sum of the necessary diagonal movements from each square unit of an n X n+1 rectangle to reach any of the corners of the rectangle.

0, 2, 16, 32, 76, 114, 204, 276, 428, 542, 772, 940, 1264, 1494, 1928, 2232, 2792, 3178, 3880, 4360, 5220, 5802, 6836, 7532, 8756, 9574, 11004, 11956, 13608, 14702, 16592, 17840, 19984, 21394, 23808, 25392, 28092, 29858, 32860, 34820, 38140, 40302, 43956

Offset: 1

Author

Isaac S. Friedman, Dec 03 2016

Keywords

• nonn

Comments

All terms in the sequence are even, because the rectangles are symmetric. A single move consists of a movement by one row and one column.

Examples

			a(3) = (13/24)(3^3) + ((3*(3 mod 2)+1)/8)*(3^2) - ((28-9*(3 mod 2))/24)*(3) - (3 mod 4)/4 = (13/24)(3^3) + (1/2)(3^2) - (19/24)(3) - (3/4) = 16.
Illustration of a(3):
.
.       3 columns
.     +---+---+---+
.   4 | 0 | 3 | 0 |  0 + 3 + 0 = 3
.     +---+---+---+
.   r | 2 | 1 | 2 |  2 + 1 + 2 = 5
.   o +---+---+---+
.   w | 2 | 1 | 2 |  2 + 1 + 2 = 5
.   s +---+---+---+
.     | 0 | 3 | 0 |  0 + 3 + 0 = 3
.     +---+---+---+
.
Adding the sums for the rows, a(3) = 3 + 5 + 5 + 3 = 16.
		

Links

• Isaac S. Friedman, Table of n, a(n) for n = 1..998
• Isaac S. Friedman, Java program to find a single term

Programs

Java

See Friedman link

Mathematica

CoefficientList[ Series[( 2(x + 7x^2 + 6x^3 + 8x^4 + 3x^5 + x^6))/((x -1)^4 (x + 1)^3 (x^2 +1)), {x, 0, 45}], x] (* or *)
LinearRecurrence[{1, 2, -2, 0, 0, -2, 2, 1, -1}, {0, 2, 16, 32, 76, 114, 204, 276, 428}, 45] (* Robert G. Wilson v, Dec 13 2016 *)

Formula

Empirical g.f.: 2*x^2*(1 + 7*x + 6*x^2 + 8*x^3 + 3*x^4 + x^5) / ((1 - x)^4*(1 + x)^3*(1 + x^2)). - Colin Barker, Dec 04 2016

Empirical: a(n) = (13/24)*(n^3) + ((3*(n mod 2) + 1)/8)*(n^2) - ((28 - 9*(n mod 2))/24)*(n) - (n mod 4)/4.