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 1-5 of 5 results.

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.

A355839 Number of vertices 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

5, 25, 133, 357, 1013, 1637, 3761, 5561, 9313, 13065, 21689, 25357, 41553, 50157, 66005, 84897, 117793, 129841, 181717, 198857, 251189, 302293, 383161, 401073, 517193, 587041, 687765, 763425, 949869, 966249, 1234425, 1320913, 1512233, 1703657, 1912765, 2023569, 2475361, 2649813, 2934997
Offset: 1

Views

Author

Scott R. Shannon, Jul 18 2022

Keywords

Comments

This sequence is similar to A355799 but here the corner vertices of the square are also connected to points on the opposite edge.

Links

Crossrefs

Cf. A355838 (regions), A355840 (edges), A355841 (k-gons), A355799 (without corner vertices), A290131, A331452, A335678.

Formula

a(n) = A355840(n) - A355838(n) + 1 by Euler's formula.

A355841 Irregular table read by rows: T(n,k) is the number of k-sided polygons formed, for k>=3, in a square when straight line segments connect 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, 128, 44, 12, 320, 152, 24, 616, 512, 84, 28, 1240, 744, 120, 40, 1936, 1928, 372, 136, 8, 3288, 2656, 616, 160, 4960, 4500, 1020, 332, 48, 7224, 6472, 1424, 392, 16, 9760, 11064, 2564, 824, 72, 16, 14144, 12424, 2696, 856, 32, 18312, 20604, 5308, 1468, 328, 16, 24384, 25392, 5968, 1584, 160, 8
Offset: 1

Views

Author

Scott R. Shannon, Jul 18 2022

Keywords

Comments

This sequence is similar to A355801 but here the corner vertices of the square are also connected to points on the opposite edge.
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 A355838 for more images of the square.

Examples

			The table begins:
4;
40;
128,   44,    12;
320,   152,   24;
616,   512,   84,    28;
1240,  744,   120,   40;
1936,  1928,  372,   136,  8;
3288,  2656,  616,   160;
4960,  4500,  1020,  332,  48;
7224,  6472,  1424,  392,  16;
9760,  11064, 2564,  824,  72,  16;
14144, 12424, 2696,  856,  32;
18312, 20604, 5308,  1468, 328, 16;
24384, 25392, 5968,  1584, 160, 8;
31816, 32768, 7564,  2652, 240, 16;
40456, 42240, 10384, 3064, 248, 24;
49384, 59152, 15068, 4680, 704, 64;
.
.

Links

Crossrefs

Cf. A355838 (regions), A355839 (vertices), A355840 (edges), A355801 (without corner vertices), A290131, A331452, A335678.

A355948 Number of edges formed in a square by straight line segments when connecting the four corner vertices to the points dividing the sides into n equal parts.

Original entry on oeis.org

8, 56, 176, 344, 632, 840, 1376, 1736, 2312, 2840, 3728, 3968, 5336, 5960, 6816, 7880, 9416, 9912, 11888, 12344, 14008, 15656, 17696, 17872, 20648, 22232, 23984, 25304, 28568, 28376, 32768, 33800, 36296, 38840, 40848, 42008, 47096, 48872, 51296, 52568, 58088, 58136, 64016, 65144, 67712, 72392
Offset: 1

Views

Author

Scott R. Shannon, Jul 21 2022

Keywords

Comments

See A108914 for images of the squares.

Crossrefs

Cf. A108914 (regions), A355949 (vertices), A355992 (k-gons), A355840, A331452, A335678.

Formula

a(n) = A108914(n) + A355949(n) - 1 by Euler's formula.

A357061 Number of edges 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

4, 12, 36, 76, 132, 180, 292, 348, 516, 604, 804, 892, 1156, 1284, 1572, 1708, 2052, 2180, 2596, 2796, 3204, 3412, 3876, 4012, 4612, 4860, 5412, 5668, 6276, 6508, 7204, 7460, 8172, 8524, 9252, 9516, 10372, 10740, 11532, 11900, 12804, 13100, 14116, 14532, 15468, 15940, 16932, 17196, 18436
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, ... .
See A357058 and A357060 for images of the squares.

Crossrefs

Cf. A357058 (regions), A357060 (vertices), A355948, A355840, A355800, A357008 (triangle).

Formula

a(n) = A357058(n) + A357060(n) - 1 by Euler's formula.
Conjecture: a(n) = 8*n^2 + 4 for squares that only contain simple intersections when cut by n internal squares. This is never the case for odd n >= 5.
Showing 1-5 of 5 results.

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