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 15 results. Next

A331452 Triangle read by rows: T(n,m) (n >= m >= 1) = number of regions (or cells) formed by drawing the line segments connecting any two of the 2*(m+n) perimeter points of an m X n grid of squares.

Original entry on oeis.org

4, 16, 56, 46, 142, 340, 104, 296, 608, 1120, 214, 544, 1124, 1916, 3264, 380, 892, 1714, 2820, 4510, 6264, 648, 1436, 2678, 4304, 6888, 9360, 13968, 1028, 2136, 3764, 6024, 9132, 12308, 17758, 22904, 1562, 3066, 5412, 8126, 12396, 16592, 23604, 29374, 38748, 2256, 4272, 7118, 10792, 16226, 20896, 29488, 36812, 47050, 58256
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Jan 27 2020

Keywords

Comments

Take a grid of m+1 X n+1 points. There are 2*(m+n) points on the perimeter. Join every pair of the perimeter points by a line segment. The lines do not extend outside the grid. T(m,n) is the number of regions formed by these lines, and A331453(m,n) and A331454(m,n) give the number of vertices and the number of line segments respectively.
A288187 is a similar sequence, except there every pair of the (m+1)*(n+1) points of the grid (including the interior points) are joined by line segments. The (m,1) (m>=1) and (2,2) entries here and in A288187 are the same, while all other entries are different.

Examples

			Triangle begins:
     4;
    16,   56;
    46,  142,  340;
   104,  296,  608,  1120;
   214,  544, 1124,  1916,  3264;
   380,  892, 1714,  2820,  4510,  6264;
   648, 1436, 2678,  4304,  6888,  9360, 13968;
  1028, 2136, 3764,  6024,  9132, 12308, 17758, 22904;
  1562, 3066, 5412,  8126, 12396, 16592, 23604, 29374, 38748;
  2256, 4272, 7118, 10792, 16226, 20896, 29488, 36812, 47050, 58256;
  ...

References

  • Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, Integers, Ron Graham Memorial Volume 21A (2021), #A5. Also in book, "Number Theory and Combinatorics: A Collection in Honor of the Mathematics of Ronald Graham", ed. B. M. Landman et al., De Gruyter, 2022, pp. 65-97.
  • Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, Integers, Ron Graham Memorial Volume 21A (2021), #A5. Also in book, "Number Theory and Combinatorics: A Collection in Honor of the Mathematics of Ronald Graham", ed. B. M. Landman et al., De Gruyter, 2022, pp. 65-97.

Links

Crossrefs

The first column is A306302, the main diagonal is A255011.
The second column is A331766.
See A333274 for the classification of vertices by valency.
Cf. A288187, A331453, A331454, A333286, A333287, A333288.

A331765 Number of edges formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of a 3 X (n+1) rectangular grid of points (or equally, a 2 X n grid of squares).

Original entry on oeis.org

28, 92, 240, 508, 944, 1548, 2520, 3754, 5392, 7528, 10296, 13570, 17844, 22768, 28584, 35704, 44048, 53380, 64728, 77292, 91500, 107828, 126408, 146772, 170080, 195580, 223764, 255010, 289792, 326996, 369320, 414908, 463880, 517724, 575404, 637530, 706172
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Feb 05 2020

Keywords

Comments

Triangles A331452, A331453, A331454 do not have formulas, except for column 1. The column 2 sequences, A331763, A331765, A331766, are therefore the next ones to attack.
See A331452 for other illustrations.

Links

Crossrefs

Column 2 of A331454.
Cf. A331763, A331766.

Extensions

More terms from Scott R. Shannon, Mar 11 2020
a(21) and beyond from Lars Blomberg, Apr 28 2020

A331763 Number of vertices formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of a 3 X (n+1) rectangular grid of points (or equally, a 2 X n grid of squares).

Original entry on oeis.org

13, 37, 99, 213, 401, 657, 1085, 1619, 2327, 3257, 4457, 5883, 7751, 9885, 12403, 15513, 19131, 23181, 28115, 33601, 39745, 46821, 54865, 63733, 73879, 84889, 97063, 110639, 125649, 141797, 160129, 179981, 201175, 224481, 249403, 276291, 306003, 337425
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Feb 05 2020

Keywords

Comments

Triangles A331452, A331453, A331454 do not have formulas, except for column 1. The column 2 sequences, A331763, A331765, A331766, are therefore the next ones to attack.
See A331452 for other illustrations.

Links

Crossrefs

Column 2 of A331453.
Cf. A331765, A331766, A331767.

Extensions

More terms from Scott R. Shannon, Mar 11 2020
a(21) and beyond from Lars Blomberg, Apr 28 2020

A331457 Triangle read by rows: T(n,k) = number of regions in a "frame" of size n X k (see Comments for definition).

Original entry on oeis.org

4, 16, 56, 46, 142, 208, 104, 296, 348, 496, 214, 544, 592, 752, 1016, 380, 892, 948, 1120, 1396, 1784, 648, 1436, 1508, 1692, 1980, 2380, 2984, 1028, 2136, 2292, 2488, 2788, 3200, 3816, 4656, 1562, 3066, 3384, 3592, 3904, 4328, 4956, 5808, 6968, 2256, 4272, 4796, 5016, 5340, 5776, 6416, 7280, 8452, 9944
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Jan 29 2020

Keywords

Comments

A "frame" of size n X k is formed from a grid of (n+1) X (k+1) points with the central grid of (n-3) X (k-3) points removed. If n or k is less than 3 then no points are removed, and T(n,k) = A331452(n,k). From now on we assume both n and k are >= 3.
The resulting array has an outer perimeter with 2*(n+k) points and an inner perimeter with 2*(n+k)-8 points, for a total of 4*(n+k)-8 perimeter points. The frame itself is the strip of width 1 between the inner and outer perimeters.
Now join every pair of perimeter points, both inner and outer, by a line segment, provided the line remains inside the frame. The sequence gives the number of regions in the resulting figure.
See A331776 for additional illustrations for the diagonal entries.
There is a crucial difference between frames of size nX2 and size nXk with k = 1 or k >= 3. If k != 2, all regions are either triangles or quadrilaterals, but for k=2 regions with larger numbers of sides can appear. Remember also that for k <= 2, the "frame" has no hole, and the graph has genus 0, whereas for k >= 3 there is a nontrivial hole and the graph has genus 1.

Examples

			Triangle begins:
4,
16,56,
46,142,208,
104,296,348,496,
214,544,592,752,1016
380,892,948,1120,1396,1784
648,1436,1508,1692,1980,2380,2984
1028,2136,2292,2488,2788,3200,3816,4656
1562,3066,3384,3592,3904,4328,4956,5808,6968
2256,4272,4796,5016,5340,5776,6416,7280,8452,9944

Links

Crossrefs

Cf. A332599 (triangle giving numbers of vertices) and A332600 (edges).
Cf. also A331452.
The first column is A306302, the main diagonal is A331776.

Formula

Column 1 is A306302, for which there is an explicit formula.
Column 2 is A331766, for which no formula is known.
For n >= k >= 3, T(n,k) = A332610(n,k) + A332611(n,k), both of which have explicit formulas.

Extensions

More terms from Scott R. Shannon, Mar 05 2020
a(8) corrected by Giovanni Resta, May 22 2025

A332606 Number of triangles in the graph formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of a 3 X (n+1) rectangular grid of points (or equally, a 2 X n grid of squares).

Original entry on oeis.org

14, 48, 102, 192, 326, 524, 802, 1192, 1634, 2296, 3074, 4052, 5246, 6740, 8398, 10440, 12770, 15512, 18782, 22384, 26386, 31204, 36482, 42232, 48826, 56508, 64318, 73356, 83366, 93996, 106010, 118788, 132634, 148600, 164814, 182648, 201998, 223172, 245634
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Mar 12 2020

Keywords

Comments

See A331452 (the illustrations for T(n,2)) for pictures of these graphs.

Links

Crossrefs

Cf. A331452, A331453, A331454, A331763, A331765, A331766, A332599, A332600, A331457, A332607, A332608, A332609.

Extensions

a(21) and beyond from Lars Blomberg, Apr 28 2020

A332607 Number of quadrilaterals in the graph formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of a 3 X (n+1) rectangular grid of points (or equally, a 2 X n grid of squares).

Original entry on oeis.org

2, 8, 36, 92, 194, 336, 554, 812, 1314, 1756, 2508, 3252, 4348, 5464, 7054, 8760, 11050, 13324, 16162, 19256, 23188, 27120, 32098, 37396, 43456, 49516, 57608, 65440, 74670, 84388, 95674, 107656, 120990, 133996, 150144, 166424, 185090, 203960, 224926, 247120
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Mar 12 2020

Keywords

Comments

See A331452 (the illustrations for T(n,2)) for pictures of these graphs.

Links

Crossrefs

Cf. A331452, A331453, A331454, A331763, A331765, A331766, A332599, A332600, A331457, A332606, A332608, A332609.

Extensions

a(21) and beyond from Lars Blomberg, Apr 28 2020

A333279 Column 2 of triangle in A288187.

Original entry on oeis.org

16, 56, 176, 388, 822, 1452, 2516, 3952, 6060, 8736, 12492, 17040, 23102, 30280, 39234, 49688, 62730, 77556, 95642, 115992, 139874, 166560, 197992, 232600, 272574, 316460, 366390, 420792, 482748, 549516, 624962, 706436, 796766, 893844, 1001074, 1115428
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Mar 15 2020

Keywords

Comments

For the graphs defined in A331452 and A288187 only the counts for graphs that are one square wide have formulas for regions, edges, and vertices (see A306302, A331757, A331755). For width 2 there are six such sequences (A331766, A331765, A331763; A333279, A333280, A333281). It would be nice to have a formula for any one of them.
The maximum number of edges over all chambers is 4 for 1 <= n <= 4 and 5 for 5 <= n <= 160. - Lars Blomberg, May 23 2021

Links

Crossrefs

Cf. A331452, A288187; A331766, A331765, A331763; A333279, A333280, A333281.

Extensions

a(10) and beyond from Lars Blomberg, May 23 2021

A333280 Column 2 of triangle in A333278.

Original entry on oeis.org

28, 92, 296, 652, 1408, 2470, 4312, 6774, 10428, 14992, 21492, 29328, 39876, 52184, 67616, 85588, 108192, 133674, 164992, 200158, 241560, 287428, 341768, 401472, 470764, 546230, 632404, 726170, 833420, 948550, 1079204, 1220054, 1376552, 1543742, 1729000
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Mar 15 2020

Keywords

Comments

For the graphs defined in A331452 and A288187 only the counts for graphs that are one square wide have formulas for regions, edges, and vertices (see A306302, A331757, A331755). For width 2 there are six such sequences (A331766, A331765, A331763; A333279, A333280, A333281). It would be nice to have a formula for any one of them.
See A333279 for illustrations.

Links

Crossrefs

Cf. A331452, A288187; A331766, A331765, A331763; A333279, A333280, A333281.

Extensions

a(10) and beyond from Lars Blomberg, May 23 2021

A333281 Column 2 of triangle in A288180.

Original entry on oeis.org

13, 37, 121, 265, 587, 1019, 1797, 2823, 4369, 6257, 9001, 12289, 16775, 21905, 28383, 35901, 45463, 56119, 69351, 84167, 101687, 120869, 143777, 168873, 198191, 229771, 266015, 305379, 350673, 399035, 454243, 513619, 579787, 649899, 727927, 810907, 903581
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Mar 15 2020

Keywords

Comments

For the graphs defined in A331452 and A288187 only the counts for graphs that are one square wide have formulas for regions, edges, and vertices (see A306302, A331757, A331755). For width 2 there are six such sequences (A331766, A331765, A331763; A333279, A333280, A333281). It would be nice to have a formula for any one of them.
See A333279 for illustrations.

Links

Crossrefs

Cf. A331452, A288187; A331766, A331765, A331763; A333279, A333280, A333281.

Extensions

a(10) and beyond from Lars Blomberg, May 23 2021

A332608 Number of pentagons in the graph formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of a 3 X (n+1) rectangular grid of points (or equally, a 2 X n grid of squares).

Original entry on oeis.org

0, 0, 4, 12, 24, 28, 80, 128, 112, 200, 236, 356, 472, 652, 656, 940, 1040, 1300, 1600, 1948, 2048, 2588, 2856, 3260, 3716, 4492, 4572, 5324, 5904, 6508, 7200, 8144, 8664, 10296, 10548, 11664, 12580, 13860, 14596, 15980, 17312, 18516, 19692, 22152, 22912
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Mar 12 2020

Keywords

Comments

See A331452 (the illustrations for T(n,2)) for pictures of these graphs.

Links

Crossrefs

Cf. A331452, A331453, A331454, A331763, A331765, A331766, A332599, A332600, A331457, A332606, A332607, A332609.

Extensions

a(21) and beyond from Lars Blomberg, Apr 28 2020
Showing 1-10 of 15 results. Next

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

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