A330850 -id:A330850 - cp's OEIS frontend

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

64, 104, 176, 304, 492, 778, 1176, 1732, 2446, 3416, 4614, 6172, 8060, 10340, 13052, 16388, 20228, 24852, 30134, 36206, 43076, 51092, 60010, 70186, 81498, 94180, 108140, 123938, 141074, 160308, 181320, 204328, 229288, 256574, 285856, 318124, 352838, 390338

Offset: 2

Scott R. Shannon and N. J. A. Sloane, Jan 28 2020

nonn

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.)
There are n+2 squares in all. The number of vertices is 3*n+2.
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.

Lars Blomberg, Table of n, a(n) for n = 2..50
Scott R. Shannon, Illustration for cross of height 2.
Scott R. Shannon, Illustration for cross of height 3.
Scott R. Shannon, Illustration for cross of height 4.
Scott R. Shannon, Illustration for cross of height 5.
Scott R. Shannon, Illustration for cross of height 6.
Scott R. Shannon, Illustration for cross of height 9.
Scott R. Shannon, Illustration for cross of height 3 using random distance-based coloring.
Scott R. Shannon, Illustration for cross of height 4 using random distance-based coloring.
Scott R. Shannon, Illustration for cross of height 5 using random distance-based coloring.
Scott R. Shannon, Illustration for cross of height 6 using random distance-based coloring.
Scott R. Shannon, Illustration for cross of height 7 using random distance-based coloring.
Scott R. Shannon, Colored illustration for a different-shaped cross, with arms of lengths 2,2,4. (There are 21858 regions.)
N. J. A. Sloane, Illustration for cross of height 2.
N. J. A. Sloane, Illustration for cross of height 3. (One of the "arms" has been cropped by the scanner, but all four arms are the same.)
N. J. A. Sloane, Illustration for cross of height 4.
N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.

Cf. A330848 (n-gons), A330850 (vertices), A330851 (edges).
See A331456 for crosses in which the arms have equal length.
A331452 is a similar sequence for a rectangular region; A007678 for a polygonal region.

a(11) and beyond from Lars Blomberg, May 31 2020

A330848 Table read by rows: T(n,k) = number of k-sided polygons in a cross with width 3 and height n (see Comments in A331455 for definition) for k = 3,4,5.

Original entry on oeis.org

50, 14, 0, 84, 20, 0, 132, 42, 2, 188, 116, 0, 290, 202, 0, 428, 350, 0, 634, 542, 0, 890, 840, 2, 1290, 1156, 0, 1776, 1640, 0, 2392, 2220, 2, 3108, 3064, 0, 4112, 3946, 2, 5260, 5076, 4, 6696, 6350, 6, 8308, 8072, 8, 10334, 9886, 8, 12588, 12256, 8

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 07 2020

See the links in A331455 for images of the crosses.

			Table begins:
50, 14, 0
84, 20, 0
132, 42, 2
188, 116, 0
290, 202, 0
428, 350, 0
634, 542, 0
890, 840, 2
1290, 1156, 0
The row sums are A331455.

Lars Blomberg, Table of n, a(n) for n = 1..147 (the first 49 rows)

Cf. A331455 (regions), A330850 (vertices), A330851 (edges), A333037 (cross with equal length arms).

a(28) and beyond from Lars Blomberg, May 31 2020

A330851 The number of edges inside a cross with width 3 and height n (see Comments in A331455 for definition) formed by the straight line segments mutually connecting all vertices and all points.

Original entry on oeis.org

108, 172, 294, 522, 848, 1352, 2046, 3032, 4260, 5958, 8048, 10806, 14082, 18070, 22778, 28646, 35314, 43436, 52676, 63370, 75290, 89330, 104908, 122816, 142476, 164644, 188954, 216686, 246584, 280386, 317186, 357658, 401138, 448884, 500014, 556706, 617160

Offset: 2

Scott R. Shannon and N. J. A. Sloane, Mar 07 2020

nonn

See the links in A331455 for images of the crosses.

Lars Blomberg, Table of n, a(n) for n = 2..50

Cf. A331455 (regions), A330848 (n-gons), A330850 (vertices), A333035 (cross with equal length arms).

Corrected offset and a(11) and beyond from Lars Blomberg, May 31 2020

A333459 Number of vertices in an diagonal-edged (or diamond-shaped) checkerboard with width and height 2*n-1 (see Comments in A333434 for definition).

Original entry on oeis.org

5, 69, 785, 4177, 12681, 33601, 71777, 125389, 238801, 379793, 548293, 878397, 1203101, 1673693, 2311477, 3063609, 3934365, 4931077, 6501545, 7873689, 9852941, 12071685, 14067849, 17728005, 20417893, 24047817, 29003717

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 22 2020

See A333434 for the definition and images of the checkerboard.

Cf. A333434 (regions), A333458 (n-gons), A333460 (edges), A333035, A333031, A330850.

a(8)-a(27) from Lars Blomberg, Jun 03 2020

cp's OEIS Frontend

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

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A330848 Table read by rows: T(n,k) = number of k-sided polygons in a cross with width 3 and height n (see Comments in A331455 for definition) for k = 3,4,5.

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A330851 The number of edges inside a cross with width 3 and height n (see Comments in A331455 for definition) formed by the straight line segments mutually connecting all vertices and all points.

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A333459 Number of vertices in an diagonal-edged (or diamond-shaped) checkerboard with width and height 2*n-1 (see Comments in A333434 for definition).

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