cp's OEIS Frontend

A331455 Number of regions in a "cross" of width 3 and height n (see Comments for definition).

%I A331455 #58 May 31 2020 05:50:34
%S A331455 64,104,176,304,492,778,1176,1732,2446,3416,4614,6172,8060,10340,
%T A331455 13052,16388,20228,24852,30134,36206,43076,51092,60010,70186,81498,
%U A331455 94180,108140,123938,141074,160308,181320,204328,229288,256574,285856,318124,352838,390338
%N A331455 Number of regions in a "cross" of width 3 and height n (see Comments for definition).
%C A331455 This "cross" of height n consists of a vertical column of n >= 2 squares with two additional squares extending to the left and right of the second square. (See illustrations.)
%C A331455 There are n+2 squares in all. The number of vertices is 3*n+2.
%C A331455 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.
%H A331455 Lars Blomberg, <a href="/A331455/b331455.txt">Table of n, a(n) for n = 2..50</a>
%H A331455 Scott R. Shannon, <a href="/A331455/a331455_6.png">Illustration for cross of height 2</a>.
%H A331455 Scott R. Shannon, <a href="/A331455/a331455_7.png">Illustration for cross of height 3</a>.
%H A331455 Scott R. Shannon, <a href="/A331455/a331455_8.png">Illustration for cross of height 4</a>.
%H A331455 Scott R. Shannon, <a href="/A331455/a331455_9.png">Illustration for cross of height 5</a>.
%H A331455 Scott R. Shannon, <a href="/A331455/a331455_10.png">Illustration for cross of height 6</a>.
%H A331455 Scott R. Shannon, <a href="/A331455/a331455_11.png">Illustration for cross of height 9</a>.
%H A331455 Scott R. Shannon, <a href="/A331455/a331455.png">Illustration for cross of height 3 using random distance-based coloring</a>.
%H A331455 Scott R. Shannon, <a href="/A331455/a331455_1.png">Illustration for cross of height 4 using random distance-based coloring</a>.
%H A331455 Scott R. Shannon, <a href="/A331455/a331455_2.png">Illustration for cross of height 5 using random distance-based coloring</a>.
%H A331455 Scott R. Shannon, <a href="/A331455/a331455_3.png">Illustration for cross of height 6 using random distance-based coloring</a>.
%H A331455 Scott R. Shannon, <a href="/A331455/a331455_4.png">Illustration for cross of height 7 using random distance-based coloring</a>.
%H A331455 Scott R. Shannon, <a href="/A331455/a331455_5.png">Colored illustration for a different-shaped cross, with arms of lengths 2,2,4.</a> (There are 21858 regions.)
%H A331455 N. J. A. Sloane, <a href="/A331455/a331455.pdf">Illustration for cross of height 2.</a>
%H A331455 N. J. A. Sloane, <a href="/A331455/a331455_1.pdf">Illustration for cross of height 3.</a> (One of the "arms" has been cropped by the scanner, but all four arms are the same.)
%H A331455 N. J. A. Sloane, <a href="/A331455/a331455_2.pdf">Illustration for cross of height 4.</a>
%H A331455 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 A331455 Cf. A330848 (n-gons), A330850 (vertices), A330851 (edges).
%Y A331455 See A331456 for crosses in which the arms have equal length.
%Y A331455 A331452 is a similar sequence for a rectangular region; A007678 for a polygonal region.
%K A331455 nonn
%O A331455 2,1
%A A331455 _Scott R. Shannon_ and _N. J. A. Sloane_, Jan 28 2020
%E A331455 a(11) and beyond from _Lars Blomberg_, May 31 2020