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
Keywords
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
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Java
See Friedman link
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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.
Comments