cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 14 results. Next

A108914 Number of regions formed inside square by diagonals and the segments joining the vertices to the points dividing the sides into n equal length segments.

Original entry on oeis.org

4, 32, 96, 188, 332, 460, 712, 916, 1204, 1488, 1904, 2108, 2716, 3080, 3532, 4068, 4772, 5140, 6016, 6392, 7188, 7992, 8936, 9260, 10484, 11312, 12208, 12968, 14396, 14660, 16504, 17220, 18436, 19680, 20756, 21548, 23692, 24728, 25992, 26868, 29204, 29704, 32176, 33068, 34444, 36552, 38552
Offset: 1

Views

Author

Len Smiley and Brian Wick ( mathclub(AT)math.uaa.alaska.edu ), Jul 19 2005

Keywords

Links

Crossrefs

A092098 is the corresponding count for triangles.
A355949 (vertices), A355948 (edges), A355992 (k-gons), A355838, A355798.

Formula

If n=1 or n is prime, a(n)=18*n^2-26*n+12.
If n is composite, vanishing regions from 3- and 4-fold concurrency must be subtracted.
a(n) = A355948(n) - A355949(n) + 1 by Euler's formula.

Extensions

a(23), a(33) corrected, a(41) and above by Scott R. Shannon, Jul 22 2022

A355799 Number of vertices formed in a square by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the square.

Original entry on oeis.org

4, 9, 25, 93, 277, 597, 1405, 2421, 4357, 6661, 11261, 14593, 23625, 30121, 41453, 54477, 75985, 87677, 122433, 139461, 177965, 216017, 275733, 298805, 383497, 439909, 522473, 588597, 729501, 763149, 963573, 1045701, 1204481, 1361789, 1546309, 1657125, 2009113, 2166617, 2418733, 2602789
Offset: 1

Views

Author

Scott R. Shannon, Jul 17 2022

Keywords

Links

Crossrefs

Cf. A355798 (regions), A355800 (edges), A355801 (k-gons), A255011 (all vertices), A290131, A331452, A335678.

Formula

a(n) = A355800(n) - A355798(n) + 1 by Euler's formula.

A355838 Number of regions formed in a square by straight line segments when connecting the n+1 points along each edge that divide it into n equal parts to the n+1 points on the edge on the opposite side of the square.

Original entry on oeis.org

4, 40, 184, 496, 1240, 2144, 4380, 6720, 10860, 15528, 24300, 30152, 46036, 57496, 75056, 96416, 129052, 148512, 198392, 225240, 279576, 336272, 415988, 453376, 565052, 648008, 754808, 848664, 1026040, 1085536, 1331532, 1452704, 1652684, 1862600, 2084888, 2247568, 2662092, 2887944, 3193744
Offset: 1

Views

Author

Scott R. Shannon, Jul 18 2022

Keywords

Comments

This sequence is similar to A355798 but here the corner vertices of the square are also connected to points on the opposite edge.

Links

Crossrefs

Cf. A355839 (vertices), A355840 (edges), A355841 (k-gons), A355798 (without corner vertices), A290131, A331452, A335678.

Formula

a(n) = A355840(n) - A355839(n) + 1 by Euler's formula.

A357058 Number of regions in a square when n internal squares are drawn between the 4n points that divide each side into n+1 equal parts.

Original entry on oeis.org

1, 5, 17, 37, 65, 93, 145, 181, 257, 309, 401, 457, 577, 653, 785, 869, 1025, 1109, 1297, 1413, 1601, 1725, 1937, 2041, 2305, 2453, 2705, 2861, 3137, 3289, 3601, 3765, 4089, 4293, 4625, 4801, 5185, 5405, 5769, 5993, 6401, 6605, 7057, 7309, 7737, 8013, 8465, 8673, 9217, 9477, 9993, 10309
Offset: 0

Views

Author

Scott R. Shannon, Sep 10 2022

Keywords

Comments

The even values of n that yield squares with non-simple intersections are 32, 38, 44, 50, 54, 62, 76, 90, 98, ... .

Links

Crossrefs

Cf. A357060 (vertices), A357061 (edges), A108914, A355838, A355798, A356984 (triangle).

Formula

a(n) = A357061(n) - A357060 (n) + 1 by Euler's formula.
Conjecture: a(n) = 4*n^2 + 1 for squares that only contain simple intersections when cut by n internal squares. This is never the case for odd n >= 5.

A355800 Number of edges formed in a square by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the square.

Original entry on oeis.org

4, 12, 48, 196, 592, 1308, 2992, 5236, 9296, 14332, 23704, 31432, 49592, 64208, 87712, 115524, 158776, 186660, 255464, 295532, 374200, 455064, 574024, 632836, 800568, 923764, 1092672, 1238412, 1515912, 1613148, 2001200, 2191124, 2516016, 2847668, 3223968, 3485484, 4167304, 4523992, 5042336
Offset: 1

Views

Author

Scott R. Shannon, Jul 17 2022

Keywords

Comments

See A355798 for images of the squares.

Crossrefs

Cf. A355798 (regions), A355799 (vertices), A355801 (k-gons), A255011 (all vertices), A290131, A331452, A335678.

Formula

a(n) = A355798(n) + A355799(n) - 1 by Euler's formula.

A355801 Irregular table read by rows: T(n,k) is the number of k-sided polygons, for k>=3, in a square when straight line segments connect the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the square.

Original entry on oeis.org

0, 1, 0, 4, 12, 12, 56, 32, 16, 156, 124, 24, 8, 0, 4, 384, 228, 72, 28, 716, 648, 144, 68, 8, 4, 1312, 1144, 240, 112, 8, 2244, 1912, 528, 256, 3528, 3072, 696, 360, 16, 5012, 5536, 1296, 524, 48, 28, 7696, 6596, 1960, 572, 16, 10340, 11448, 2968, 1028, 160, 24, 14520, 14428, 3872, 1156, 104, 8
Offset: 1

Views

Author

Scott R. Shannon, Jul 17 2022

Keywords

Comments

Up to n = 50 the maximum sided k-gon created is the 8-gon. It is plausible this is the maximum sided k-gon for all n, although this is unknown.
See A355798 for more images of the square.
The keyword "look" is for the n = 10 image. - N. J. A. Sloane, Jul 21 2022

Examples

			The table begins:
0,     1;
0,     4;
12,    12;
56,    32,    16;
156,   124,   24,   8,    0,   4;
384,   228,   72,   28;
716,   648,   144,  68,   8,   4;
1312,  1144,  240,  112,  8;
2244,  1912,  528,  256;
3528,  3072,  696,  360,  16;
5012,  5536,  1296, 524,  48,  28;
7696,  6596,  1960, 572,  16;
10340, 11448, 2968, 1028, 160, 24;
14520, 14428, 3872, 1156, 104, 8;
19588, 19156, 5296, 2052, 160, 8;
25392, 26112, 7160, 2152, 208, 24;
31820, 37244, 9936, 3240, 488, 64;
.
.

Links

Crossrefs

Cf. A355798 (regions), A355799 (vertices), A355800 (edges), A355801 (k-gons), A255011 (all vertices), A290131, A331452, A335678.

A358407 Number of regions formed in a square by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on each of the two adjacent edges of the square.

Original entry on oeis.org

1, 5, 37, 173, 553, 1365, 2909, 5513, 9577, 15485, 24157, 35021, 51201, 71013, 95621, 126277, 167213, 211737, 272025, 335681, 413677, 505445, 618557, 729485, 878017, 1034697, 1215185, 1409273, 1654785, 1875265, 2192281, 2486797, 2836317, 3216833, 3633709, 4034313, 4599789, 5124841
Offset: 1

Views

Author

Scott R. Shannon, Nov 14 2022

Keywords

Links

Crossrefs

Cf. A358408 (vertices), A358409 (edges), A355798, A255011.

Formula

a(n) = A358409(n) - A358408(n) + 1 by Euler's formula.

A359653 Number of regions formed in a square with edge length 1 by straight line segments when connecting the internal edge points that divide the sides into segments with lengths equal to the Farey series of order n to the equivalent points on the opposite side of the square.

Original entry on oeis.org

1, 4, 96, 728, 7840, 17744, 104136, 246108, 681704, 1187200, 3719496, 5396692, 14149896
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Jan 09 2023

Keywords

Comments

The number of points internal to each edge is given by A005728(n) - 2.

Links

Crossrefs

Cf. A359654 (vertices), A359655 (edges), A359656 (k-gons), A005728, A358886, A358882, A355798, A358948, A006842, A006843.

Formula

a(n) = A359655(n) - A359654(n) + 1 by Euler's formula.

A367662 Number of regions formed in a hexagon by straight line segments when connecting the n-1 points between each corner that divide each edge into n equal parts to the n-1 points on the edge on the opposite side of the hexagon.

Original entry on oeis.org

1, 6, 54, 252, 780, 1968, 4146, 7662, 13014, 21078, 32562, 46818, 67962, 93048, 123684, 166794, 217380, 271056, 349932, 428250, 523272, 645744, 784794, 918606, 1104966, 1304250, 1515018, 1768974, 2071662, 2317602, 2734242, 3101670, 3486990, 3987774, 4460862, 4978530, 5688804, 6316332
Offset: 1

Views

Author

Scott R. Shannon, Nov 26 2023

Keywords

Comments

Keyword "look" is because of the linked images. - N. J. A. Sloane, Dec 01 2023

Links

Crossrefs

Cf. A367663 (vertices), A367664 (edges), A367665 (k-gons), A355798.

Formula

a(n) = A367664(n) - A367663(n) + 1 by Euler's formula.

A356790 Table read by antidiagonals: T(n,k) (n >= 1, k >= 1) is the number of regions formed by straight line segments when connecting the k-1 points along the top side of a rectangle to each of the k-1 points along the bottom side that divide these sides into k equal parts, along with straight lines that directly connect the n-1 points along the left side to the diametrically opposite point on the right side that divide these sides into n equal parts.

Original entry on oeis.org

1, 2, 2, 6, 4, 3, 18, 10, 6, 4, 48, 24, 16, 8, 5, 106, 56, 34, 20, 10, 6, 216, 116, 70, 44, 26, 12, 7, 382, 228, 134, 84, 58, 30, 14, 8, 650, 396, 250, 152, 112, 60, 36, 16, 9, 1030, 666, 422, 272, 190, 112, 78, 40, 18, 10, 1564, 1048, 696, 448, 320, 196, 150, 84, 46, 20, 11
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Sep 04 2022

Keywords

Examples

			The table begins:
1,  2,  6,  18,  48,  106, 216, 382, 650,  1030, 1564, 2258, 3210, 4386, 5926, ...
2,  4,  10, 24,  56,  116, 228, 396, 666,  1048, 1584, 2280, 3234, 4412, 5954, ...
3,  6,  16, 34,  70,  134, 250, 422, 696,  1082, 1622, 2322, 3280, 4462, 6008, ...
4,  8,  20, 44,  84,  152, 272, 448, 726,  1116, 1660, 2364, 3326, 4512, 6062, ...
5,  10, 26, 58,  112, 190, 320, 506, 794,  1194, 1748, 2462, 3434, 4630, 6190, ...
6,  12, 30, 60,  112, 196, 326, 512, 800,  1200, 1754, 2468, 3440, 4636, 6196, ...
7,  14, 36, 78,  150, 258, 418, 626, 936,  1358, 1934, 2670, 3664, 4882, 6464, ...
8,  16, 40, 84,  152, 256, 414, 632, 942,  1364, 1940, 2676, 3670, 4888, 6470, ...
9,  18, 46, 94,  172, 290, 468, 710, 1050, 1490, 2084, 2838, 3850, 5086, 6686, ...
10, 20, 50, 104, 188, 304, 480, 720, 1060, 1516, 2112, 2868, 3882, 5120, 6722, ...
11, 22, 56, 118, 218, 366, 586, 878, 1280, 1794, 2454, 3258, 4320, 5606, 7256, ...
12, 24, 60, 120, 208, 336, 518, 764, 1114, 1580, 2204, 2992, 4020, 5272, 6888, ...
.
.
See the attached table for further terms.

Links

Crossrefs

Cf. A146951, A355798, A306302, A290131, A331452, A355902.

Formula

T(1,k) = A306302(k-2) + 2, k >= 2.
T(2,k) = 2*A355902(k-2) + 4 = A306302(k-2) + 2*k, k >= 2.
T(n,1) = n.
T(n,2) = 2n.
T(n,3) = A146951(n).
Showing 1-10 of 14 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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