A332599 -id:A332599 - cp's OEIS frontend

A332600 Triangle read by rows: T(n,k) = number of edges in a "frame" of size n X k (see Comments in A331457 for definition).

8, 28, 92, 80, 240, 360, 178, 508, 604, 860, 372, 944, 1040, 1320, 1792, 654, 1548, 1652, 1956, 2452, 3124, 1124, 2520, 2640, 2968, 3488, 4184, 5256, 1782, 3754, 4004, 4356, 4900, 5620, 6716, 8188, 2724, 5392, 5936, 6312, 6880, 7624, 8744, 10240, 12304, 3914, 7528, 8364, 8764, 9356, 10124, 11268, 12788, 14876, 17460

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 03 2020

See A331457 and A331776 for further illustrations.

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.

			Triangle begins:
[8],
[28, 92],
[80, 240, 360],
[178, 508, 604, 860],
[372, 944, 1040, 1320, 1792],
[654, 1548, 1652, 1956, 2452, 3124],
[1124, 2520, 2640, 2968, 3488, 4184, 5256],
[1782, 3754, 4004, 4356, 4900, 5620, 6716, 8188],
[2724, 5392, 5936, 6312, 6880, 7624, 8744, 10240, 12304],
[3914, 7528, 8364, 8764, 9356, 10124, 11268, 12788, 14876, 17460],
...

Scott R. Shannon, Colored illustration for T(3,3) = 360
Scott R. Shannon, Colored illustration for T(4,4) = 860
N. J. A. Sloane, Illustration for T(3,3) = 360.

Cf. A331457, A331757, A331765, A331776, A332599, A332610, A332611.

The main diagonal is A332597.

Column 1 is A331757, for which there is an explicit formula.
Column 2 is A331765, for which no formula is known.
For m >= n >= 3, T(m,n) = (3*A332610(m,n)+4*A332611(m,n)+4*m+4*n-8)/2, and both A332610 and A332611 have explicit formulas.

More terms from N. J. A. Sloane, Mar 13 2020

A330846 The number of vertices inside a hexagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Original entry on oeis.org

19, 301, 1753, 6913, 17713, 38497, 80473, 139927, 225595, 356329, 549967, 756901, 1113283, 1474867, 1913365, 2620789, 3418081, 4119175, 5428993, 6453235, 7861957, 9761335, 11913115, 13695055, 16489633, 19439797, 22464883, 25997329, 30860593, 33866305

Offset: 1

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

nonn

See the links in A331931 for images of the hexagons.

Cf. A331931 (regions), A331932 (n-gons), A330845 (edges), A092866, A332599, A007569.

a(9)-a(30) from Lars Blomberg, May 12 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

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

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.

			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

Scott R. Shannon, Colored illustration for T(1,1) = 4.
Scott R. Shannon, Colored illustration for T(2,2) = 56.
Scott R. Shannon, Colored illustration for T(3,3) = 208.
Scott R. Shannon, Colored illustration for T(4,4) = 496.
Scott R. Shannon, Colored illustration for T(5,5) = 1016.
Scott R. Shannon, Colored illustration for T(6,6) = 1784.
Scott R. Shannon, Colored illustration for T(7,4) = 1692.
Scott R. Shannon, Colored illustration for T(10,6) = 5776.
N. J. A. Sloane, Illustration for T(3,3) = 208.

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

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.

More terms from Scott R. Shannon, Mar 05 2020

a(8) corrected by Giovanni Resta, May 22 2025

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

Scott R. Shannon and N. J. A. Sloane, Mar 06 2020

nonn

See the links in A331456 for images of the crosses.

Lars Blomberg, Table of n, a(n) for n = 0..48

Cf. A331456 (regions), A333036 (edges), A333037 (n-gons), A092866, A332599, A007569.

a(11) and beyond from Lars Blomberg, May 30 2020

A330847 The number of vertices inside a pentagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Original entry on oeis.org

10, 121, 1055, 3506, 10410, 21111, 43740, 74526, 124490, 190291, 288190, 403321, 573805, 771191, 1027550, 1331896, 1721765, 2157301, 2712175, 3322441, 4067325, 4904491, 5900185, 6975151, 8268755, 9678566, 11297345, 13060231, 15102955, 17263311, 19784995

Offset: 1

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

nonn

See the links in A331929 for images of the pentagons.

Lars Blomberg, Table of n, a(n) for n = 1..35

Cf. A331929 (regions), A331939 (n-gons), A329710 (edges), A330846, A092866, A332599, A007569

a(9) and beyond from Lars Blomberg, May 11 2020

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

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

nonn

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

Lars Blomberg, Table of n, a(n) for n = 1..100
Scott R. Shannon, Data specifically for nX2 (or 2Xn) rectangles

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

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

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

nonn

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

Lars Blomberg, Table of n, a(n) for n = 1..100
Scott R. Shannon, Data specifically for nX2 (or 2Xn) rectangles

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

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

A332418 The number of vertices on a decagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Original entry on oeis.org

171, 3581, 23651, 80191, 213041, 444251, 862711, 1481141, 2413721, 3701951, 5493891, 7765621, 10833601, 14589491, 19315751, 25064491, 32107771, 40337021, 50328771, 61790891, 75318371

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 09 2020

See the links in A333139 for images of the decagons.

Cf. A333139 (regions), A332417 (n-gons), A332419 (edges), A330846, A092866, A332599, A007569.

a(6)-a(21) from Lars Blomberg, May 18 2020

A332428 The number of vertices on a nonagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Original entry on oeis.org

135, 2395, 16434, 53155, 141147, 293374, 565767, 966493, 1580940, 2411533, 3581613, 5070655, 7057026, 9493435, 12594564, 16307974, 20902338, 26269597, 32760774, 40217905, 49049919, 59090671, 70803180

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 09 2020

See the links in A332421 for images of the nonagons.

Cf. A332421 (regions), A332427 (n-gons), A332429 (edges), A330846, A092866, A332599, A007569.

a(6)-a(23) from Lars Blomberg, May 16 2020

A333109 The number of vertices on an octagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Original entry on oeis.org

57, 1145, 8417, 29121, 80345, 167105, 333297, 570969, 939113, 1441153, 2153937, 3029913, 4262929, 5741473, 7606745, 9876585, 12690553, 15921777, 19922289, 24430633, 29834073, 35990065, 43151521, 51068689

Offset: 1

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

See the links in A333075 for images of the octagons.

Cf. A333075 (regions), A333076 (n-gons), A333110 (edges), A330846, A092866, A332599, A007569.

a(7)-a(24) from Lars Blomberg, May 14 2020

cp's OEIS Frontend

A332600 Triangle read by rows: T(n,k) = number of edges in a "frame" of size n X k (see Comments in A331457 for definition).

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A330846 The number of vertices inside a hexagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

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

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A333035 Number of vertices in an equal-armed cross with arms of length n (see Comments in A331456 for definition).

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A330847 The number of vertices inside a pentagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

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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).

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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).

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A332418 The number of vertices on a decagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

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A332428 The number of vertices on a nonagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

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A333109 The number of vertices on an octagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

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