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

1,2

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.

Links

Isaac S. Friedman, Table of n, a(n) for n = 1..998

Isaac S. Friedman, Java program to find a single term

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.

Example

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.

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 *)

Prog

(Java) See Friedman link

Crossrefs

Sequence in context: A056707 A069256 A344016 * A120069 A018975 A012696

Adjacent sequences: A279031 A279032 A279033 * A279035 A279036 A279037

Keyword

nonn

Author

Isaac S. Friedman, Dec 03 2016

Status

approved