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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.

%I A279034 #47 Dec 21 2016 12:47:53
%S A279034 0,2,16,32,76,114,204,276,428,542,772,940,1264,1494,1928,2232,2792,
%T A279034 3178,3880,4360,5220,5802,6836,7532,8756,9574,11004,11956,13608,14702,
%U A279034 16592,17840,19984,21394,23808,25392,28092,29858,32860,34820,38140,40302,43956
%N 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.
%C A279034 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.
%H A279034 Isaac S. Friedman, <a href="/A279034/b279034.txt">Table of n, a(n) for n = 1..998</a>
%H A279034 Isaac S. Friedman, <a href="/A279034/a279034.txt">Java program to find a single term</a>
%F A279034 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
%F A279034 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.
%e A279034 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.
%e A279034 Illustration of a(3):
%e A279034 .
%e A279034 .       3 columns
%e A279034 .     +---+---+---+
%e A279034 .   4 | 0 | 3 | 0 |  0 + 3 + 0 = 3
%e A279034 .     +---+---+---+
%e A279034 .   r | 2 | 1 | 2 |  2 + 1 + 2 = 5
%e A279034 .   o +---+---+---+
%e A279034 .   w | 2 | 1 | 2 |  2 + 1 + 2 = 5
%e A279034 .   s +---+---+---+
%e A279034 .     | 0 | 3 | 0 |  0 + 3 + 0 = 3
%e A279034 .     +---+---+---+
%e A279034 .
%e A279034 Adding the sums for the rows, a(3) = 3 + 5 + 5 + 3 = 16.
%t A279034 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 *)
%t A279034 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 *)
%o A279034 (Java) See Friedman link
%K A279034 nonn
%O A279034 1,2
%A A279034 _Isaac S. Friedman_, Dec 03 2016