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A331456 Number of regions in an equal-armed cross with arms of length n (see Comments for definition).

%I A331456 #61 Sep 19 2020 12:37:42
%S A331456 4,104,568,1900,4808,10180,19180,33132,53628,82432,121448,172948,
%T A331456 239356,323168,427272,554892,709476,893772,1111588,1367292,1664604,
%U A331456 2008240,2402560,2852532,3363280,3938712,4585568,5308720,6112736,7006068,7994412,9084788,10281812
%N A331456 Number of regions in an equal-armed cross with arms of length n (see Comments for definition).
%C A331456 This cross of height n consists of a central square with 4 arms of length n.
%C A331456 There are 4n+1 squares in all. The number of vertices is 8n+4.
%C A331456 Now join every pair of vertices by a line segment, provided the line does not extend beyond the boundary of the cross. The sequence gives the number of regions in the resulting figure.
%C A331456 See A337641 for information about these regions, their numbers of sides, their coordinates, and for further illustrations. - _N. J. A. Sloane_, Sep 17 2020
%H A331456 Lars Blomberg, <a href="/A331456/b331456.txt">Table of n, a(n) for n = 0..48</a>
%H A331456 Scott R. Shannon, <a href="/A331452/a331452_6.png">Colored illustration for a(0).</a>
%H A331456 Scott R. Shannon, <a href="/A331456/a331456.png">Colored illustration for a(1).</a>
%H A331456 Scott R. Shannon, <a href="/A331456/a331456_1.png">Colored illustration for a(2).</a>
%H A331456 Scott R. Shannon, <a href="/A331456/a331456_2.png">Colored illustration for a(3).</a>
%H A331456 Scott R. Shannon, <a href="/A331456/a331456_3.png">Colored illustration for a(4).</a>
%H A331456 Scott R. Shannon, <a href="/A331456/a331456_4.png">Colored illustration for a(5).</a>
%H A331456 Scott R. Shannon, <a href="/A331456/a331456_5.png">Colored illustration for a(9).</a>
%H A331456 Scott R. Shannon, <a href="/A331456/a331456_6.png">Colored illustration for a(1) classifying nodes and cells.</a>
%H A331456 Scott R. Shannon, <a href="/A331456/a331456_7.png">Colored illustration for a(2) classifying nodes and cells.</a>
%H A331456 Scott R. Shannon, <a href="/A331456/a331456_8.png">Colored illustration for a(3) classifying nodes and cells.</a>
%H A331456 Scott R. Shannon, <a href="/A331456/a331456_9.png">Colored illustration for a(4) classifying nodes and cells.</a>
%H A331456 Scott R. Shannon, <a href="/A331456/a331456_10.png">Colored illustration for a(5) classifying nodes and cells.</a>
%H A331456 Scott R. Shannon, <a href="/A331456/a331456_11.png">Colored illustration for a(6) classifying nodes and cells.</a>
%H A331456 N. J. A. Sloane, <a href="/A331455/a331455_1.pdf">Illustration for a(1).</a> (One of the "arms" has been cropped by the scanner, but all four arms are the same.)
%H A331456 N. J. A. Sloane (in collaboration with Scott R. Shannon), <a href="/A331452/a331452.pdf">Art and Sequences</a>, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.
%Y A331456 Cf. A333035 (vertices), A333036 (edges), A333037 (n-gons), A337641.
%Y A331456 See A331455 for a different family of crosses.
%Y A331456 A331452 is a similar sequence for a rectangular region; A007678 for a polygonal region.
%Y A331456 Cf. A331458.
%K A331456 nonn
%O A331456 0,1
%A A331456 _Scott R. Shannon_ and _N. J. A. Sloane_, Jan 28 2020
%E A331456 a(11) and beyond from _Lars Blomberg_, May 30 2020