A333035
Number of vertices in an equal-armed cross with arms of length n (see Comments in A331456 for definition).
Original entry on oeis.org
5, 69, 397, 1417, 3717, 8009, 15361, 26777, 43697, 67597, 100245, 143177, 199009, 269197, 356789, 464577, 595521, 751129, 935825, 1151881, 1403953, 1695765, 2031485, 2413337, 2848373, 3337781, 3888917, 4505277, 5191557, 5952313, 6796713, 7725945, 8747121
Offset: 0
A333036
Number of edges in an equal-armed cross with arms of length n (see Comments in A331456 for definition).
Original entry on oeis.org
8, 172, 964, 3316, 8524, 18188, 34540, 59908, 97324, 150028, 221692, 316124, 438364, 592364, 784060, 1019468, 1304996, 1644900, 2047412, 2519172, 3068556, 3704004, 4434044, 5265868, 6211652, 7276492, 8474484, 9813996, 11304292, 12958380, 14791124, 16810732
Offset: 0
A333037
Table read by rows: T(n,k) = number of k-sided polygons in an equal-armed cross with arms of length n (see Comments in A331456 for definition) for k = 3,4,5,6,7.
Original entry on oeis.org
4, 0, 0, 0, 0, 84, 20, 0, 0, 0, 380, 180, 0, 8, 0, 1076, 764, 40, 20, 0, 2380, 2316, 64, 48, 0, 4716, 5188, 224, 52, 0, 8236, 10492, 360, 92, 0, 13620, 18772, 632, 108, 0, 21188, 31380, 864, 196, 0, 31596, 49228, 1376, 224, 8, 44980, 74268, 1920, 272, 8
Offset: 0
Table begins:
4, 0, 0, 0, 0
84, 20, 0, 0, 0
380, 180, 0, 8, 0
1076, 764, 40, 20, 0
2380, 2316, 64, 48, 0
4716, 5188, 224, 52, 0
8236, 10492, 360, 92, 0
13620, 18772, 632, 108, 0
21188, 31380, 864, 196, 0
31596, 49228, 1376, 224, 8
44980, 74268, 1920, 272, 8
The row sums are A331456.
Original entry on oeis.org
1, 26, 142, 475, 1202, 2545, 4795, 8283, 13407, 20608, 30362, 43237, 59839, 80792, 106818, 138723, 177369, 223443, 277897, 341823, 416151, 502060, 600640, 713133, 840820, 984678, 1146392, 1327180, 1528184, 1751517, 1998603, 2271197, 2570453, 2898126, 3256414
Offset: 0
A337641
One-quarter of the number of regions in the central square of an equal-armed cross with arms of length n (as in A331456).
Original entry on oeis.org
1, 14, 70, 231, 576, 1207, 2255, 3883, 6267, 9588, 14088, 20021, 27667, 37306, 49240, 63859, 81517, 102603, 127545, 156769, 190739, 229932, 274898, 326181, 384332, 449878, 523472, 605766, 697380, 799053, 911449, 1035371, 1171471, 1320566, 1483374, 1660873, 1853819, 2063133, 2289607, 2534326, 2798159
Offset: 0
- Lars Blomberg, Table of n, a(n) for n = 0..74
- Scott R. Shannon, Colored illustration for a(0): there are 4 regions, so a(0) = 1.
- Scott R. Shannon, Colored illustration for a(1): there are 56 regions, so a(1) = 14.
- Scott R. Shannon, Colored illustration for a(2): there are 280 regions, so a(2) = 70.
- Scott R. Shannon, Colored illustration for a(3): there are 924 regions, so a(3) = 231.
- Scott R. Shannon, Black and white illustration for a(1) (Shows vertices and regions for each square)
- Scott R. Shannon, Black and white illustration for a(2) (Shows vertices and regions for each square)
- Scott R. Shannon, Black and white illustration for a(3) (Shows vertices and regions for each square)
- Scott R. Shannon, Black and white illustration for a(4) (Shows vertices and regions for each square)
- Scott R. Shannon, Black and white illustration for a(5) (Shows vertices and regions for each square)
- Scott R. Shannon, Black and white illustration for a(6) (Shows vertices and regions for each square)
A331455
Number of regions in a "cross" of width 3 and height n (see Comments for definition).
Original entry on oeis.org
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
- 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.
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.
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.
Original entry on oeis.org
4, 104, 1080, 5220, 15508, 39088, 81464, 144292, 261544, 415552, 610460, 942032, 1303848, 1803360, 2461232, 3250284, 4182552, 5269080, 6818764, 8326188, 10336548, 12621292, 14882600, 18368708, 21377496, 25168908, 29994204
Offset: 1
For n = 1 the board is a single square with 4 vertices on the corners.
For n = 2 the board contains 12 vertices, represented by '*', shown below:
*---*
| |
*---* *---*
| |
*---* *---*
| |
*---*
.
For n = 3 the board contains 20 vertices, represented by '*', shown below:
*---*
| |
*---* *---*
| |
*---* *---*
| |
*---* *---*
| |
*---* *---*
| |
*---*
.
- Scott R. Shannon, Illustration for n = 2.
- Scott R. Shannon, Illustration for n = 3.
- Scott R. Shannon, Illustration for n = 4.
- Scott R. Shannon, Illustration for n = 5.
- Scott R. Shannon, Illustration for n = 6.
- Scott R. Shannon, Illustration for n = 2 using random distance-based coloring.
- Scott R. Shannon, Illustration for n = 3 using random distance-based coloring.
- Scott R. Shannon, Illustration for n = 4 using random distance-based coloring.
- Scott R. Shannon, Illustration for n = 5 using random distance-based coloring.
- Scott R. Shannon, Illustration for n = 6 using random distance-based coloring.
A335861
Number of regions in a Y-shaped polygon with equal arms of length n (see the Comments for definition).
Original entry on oeis.org
1, 70, 349, 916, 1474, 2296, 3412, 4978, 7042, 9748, 13132, 17506, 22786, 29410, 37288, 46630, 57574
Offset: 0
a(0) = 1. There is one region in an equilateral triangle with no other polygons.
a(1) = 70. With one square adjacent to each of the triangles sides the resulting line segments form 48 triangles, twelves 4-gons, nine 5-gons, and one 6-gon. This gives a total of 70 regions. See the first linked image.
A337790
Number of vertices in a Y-shaped polygon with equal arms of length n (see the Comments in A335861 for definition).
Original entry on oeis.org
3, 57, 306, 837, 1335, 2073, 3033, 4395, 6147, 8469, 11253, 14907, 19263, 24819, 31197, 38823, 47619
Offset: 0
a(0) = 3. A single triangle with no other polygons has three vertices.
a(1) = 57. With one square adjacent to each of the triangles sides the resulting line segments form 51 vertices shared by four polygons, 3 vertices shared by six polygons, and 3 vertices shared by seven polygons. This gives a total of 57 vertices. See the first linked image.
Showing 1-9 of 9 results.
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