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A333434 The number of regions inside a diagonal-edged (or diamond-shaped) checkerboard of width and height 2*n-1 formed by the straight line segments mutually connecting any two of the 8*n-4 vertices on the perimeter.

%I A333434 #26 Jun 03 2020 05:27:25
%S A333434 4,104,1080,5220,15508,39088,81464,144292,261544,415552,610460,942032,
%T A333434 1303848,1803360,2461232,3250284,4182552,5269080,6818764,8326188,
%U A333434 10336548,12621292,14882600,18368708,21377496,25168908,29994204
%N A333434 The number of regions inside a diagonal-edged (or diamond-shaped) checkerboard of width and height 2*n-1 formed by the straight line segments mutually connecting any two of the 8*n-4 vertices on the perimeter.
%C A333434 The diagonal-edged checker board of width and height 2*n-1 contains 8*n-4 vertices lying on a 2D square grid as shown in the examples below. Join every pair of vertices by a line segment, provided the line does not extend beyond the boundary of the board. The sequence gives the number of regions in the resulting figure.
%H A333434 Scott R. Shannon, <a href="/A333434/a333434.png">Illustration for n = 2</a>.
%H A333434 Scott R. Shannon, <a href="/A333434/a333434_1.png">Illustration for n = 3</a>.
%H A333434 Scott R. Shannon, <a href="/A333434/a333434_2.png">Illustration for n = 4</a>.
%H A333434 Scott R. Shannon, <a href="/A333434/a333434_3.png">Illustration for n = 5</a>.
%H A333434 Scott R. Shannon, <a href="/A333434/a333434_4.png">Illustration for n = 6</a>.
%H A333434 Scott R. Shannon, <a href="/A333434/a333434_5.png">Illustration for n = 2 using random distance-based coloring</a>.
%H A333434 Scott R. Shannon, <a href="/A333434/a333434_6.png">Illustration for n = 3 using random distance-based coloring</a>.
%H A333434 Scott R. Shannon, <a href="/A333434/a333434_7.png">Illustration for n = 4 using random distance-based coloring</a>.
%H A333434 Scott R. Shannon, <a href="/A333434/a333434_10.png">Illustration for n = 5 using random distance-based coloring</a>.
%H A333434 Scott R. Shannon, <a href="/A333434/a333434_9.png">Illustration for n = 6 using random distance-based coloring</a>.
%e A333434 For n = 1 the board is a single square with 4 vertices on the corners.
%e A333434 For n = 2 the board contains 12 vertices, represented by '*', shown below:
%e A333434           *---*
%e A333434           |   |
%e A333434       *---*   *---*
%e A333434       |           |
%e A333434       *---*   *---*
%e A333434           |   |
%e A333434           *---*
%e A333434 .
%e A333434 For n = 3 the board contains 20 vertices, represented by '*', shown below:
%e A333434           *---*
%e A333434           |   |
%e A333434       *---*   *---*
%e A333434       |           |
%e A333434   *---*           *---*
%e A333434   |                   |
%e A333434   *---*           *---*
%e A333434       |           |
%e A333434       *---*   *---*
%e A333434           |   |
%e A333434           *---*
%e A333434 .
%Y A333434 Cf. A333458 (n-gons), A333459 (vertices), A333460 (edges), A331452, A331456, A331911.
%K A333434 nonn,more
%O A333434 1,1
%A A333434 _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 21 2020
%E A333434 a(8)-a(27) from _Lars Blomberg_, Jun 03 2020