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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 13 18:00 EDT 2022. Contains 356107 sequences. (Running on oeis4.)