A306302 -id:A306302 - cp's OEIS frontend

A333281 Column 2 of triangle in A288180.

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

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

nonn

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.

Lars Blomberg, Table of n, a(n) for n = 1..200
Hugo Pfoertner, Illustrations of Chamber Complexes up to 5 X 5.

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

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

A333282 Triangle read by rows: T(m,n) (m >= n >= 1) = number of regions formed by drawing the line segments connecting any two of the (m+1) X (n+1) lattice points in an m X n grid of squares and extending them to the boundary of the grid.

Original entry on oeis.org

4, 16, 56, 46, 192, 624, 104, 428, 1416, 3288, 214, 942, 3178, 7520, 16912, 380, 1672, 5612, 13188, 29588, 51864, 648, 2940, 9926, 23368, 52368, 92518, 164692, 1028, 4624, 15732, 37184, 83628, 148292, 263910, 422792, 1562, 7160, 24310, 57590, 130034, 230856, 410402, 658080, 1023416

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 16 2020

Triangle gives number of nodes in graph LC(m,n) in the notation of Blomberg-Shannon-Sloane (2020).

If we only joined pairs of the 2(m+n) boundary points, we would get A331452. If we did not extend the lines to the boundary of the grid, we would get A288187. (One of the links below shows the difference between the three definitions in the case m=3, n=2.)

			Triangle begins:
4,
16, 56,
46, 192, 624,
104, 428, 1416, 3288,
214, 942, 3178, 7520, 16912,
380, 1672, 5612, 13188, 29588, 51864,
648, 2940, 9926, 23368, 52368, 92518, 164692,
1028, 4624, 15732, 37184, 83628, 148292, 263910, 422792
1562, 7160, 24310, 57590, 130034, 230856, 410402, 658080, 1023416
2256, 10336, 35132, 83116, 187376, 331484, 588618, 942808, 1466056, 2101272

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.
Seppo Mustonen, Statistical accuracy of geometric constructions, 2008.
Seppo Mustonen, Statistical accuracy of geometric constructions, 2008 [Local copy]
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points, 2009
Seppo Mustonen, On lines and their intersection points in a rectangular grid of points, 2009 [Local copy]
Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points, 2010
Seppo Mustonen, On lines going through a given number of points in a rectangular grid of points, 2010 [Local copy]
Scott R. Shannon, Colored illustration for T(3,2)
Scott R. Shannon, Colored illustration for T(3,2) (edge number coloring)
Scott R. Shannon, Colored illustration for T(3,3)
Scott R. Shannon, Colored illustration for T(3,3) (edge number coloring)
Scott R. Shannon, Colored illustration for T(4,4)
Scott R. Shannon, Colored illustration for T(4,4) (edge number coloring)
Scott R. Shannon, Colored illustration for T(5,5)
Scott R. Shannon, Colored illustration for T(5,5) (edge number coloring)
Scott R. Shannon, Colored illustration for T(6,3)
Scott R. Shannon, Colored illustration for T(6,3) (edge number coloring)
Scott R. Shannon, Colored illustration for T(7,4)
Scott R. Shannon, Colored illustration for T(7,4) (edge number coloring)
Scott R. Shannon, Colored illustration for T(8,2)
Scott R. Shannon, Colored illustration for T(8,2) (edge number coloring)
Scott R. Shannon, Numerical properties of these structures. Corrected version Mar 24 2020.
N. J. A. Sloane, Illustration of T(3,2) = 192. [Black lines correspond to A331452(3,2), black + red lines correspond to A288187(3,2), and black + red + blue lines to T(3,2)]
N. J. A. Sloane, Illustration of T(3,3) = 624 [Black lines correspond to A288187(3,3), and black + red lines to T(3,3)]

Cf. A288187, A331452, A333283 (edges), A333284 (vertices). Column 1 is A306302. Main diagonal is A333294.

More terms and corrections from Scott R. Shannon, Mar 21 2020

More terms from Scott R. Shannon, May 27 2021

A333276 a(n) = Sum_k k*A333274(n,k).

Original entry on oeis.org

12, 50, 152, 346, 732, 1294, 2232, 3546, 5428, 7806, 11136, 15226, 20676, 27150, 35048, 44386, 56044, 69302, 85480, 103882, 125180, 148942, 176968, 208034, 243772, 283014, 327272, 375826, 431212, 490918, 558456, 631978, 712844, 799726, 895152, 997322, 1110628

Offset: 1

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

nonn

a(n)/A331755(n) is the average number of polygons touching a vertex in the graph defined in A306302.

Lars Blomberg, Table of n, a(n) for n = 1..100
Scott R. Shannon, Images of vertices for n=1.
Scott R. Shannon, Images of vertices for n=2.
Scott R. Shannon, Images of vertices for n=3.
Scott R. Shannon, Images of vertices for n=4.
Scott R. Shannon, Images of vertices for n=9.
Scott R. Shannon, Images of vertices for n=14.

Cf. A159065, A306302, A333274, A333275, A333277.

a(15) and beyond from Lars Blomberg, Jun 17 2020

A335679 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of edges in figure formed by taking m equally spaced points on a line and n equally spaced points on a parallel line, and joining each of the m points to each of the n points by a line segment.

Original entry on oeis.org

1, 3, 3, 5, 8, 5, 7, 15, 15, 7, 9, 24, 28, 24, 9, 11, 35, 47, 47, 35, 11, 13, 48, 69, 80, 69, 48, 13, 15, 63, 97, 119, 119, 97, 63, 15, 17, 80, 128, 170, 178, 170, 128, 80, 17, 19, 99, 165, 225, 257, 257, 225, 165, 99, 19, 21, 120, 205, 292, 340, 372, 340, 292, 205, 120, 21

Offset: 1

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Jun 28 2020

The case m=n (the main diagonal) is dealt with in A331757. A306302 has illustrations for the diagonal case for m = 1 to 15.

Also A335678 has colored illustrations for many values of m and n.

			The initial rows of the array are:
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, ...
3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, ...
5, 15, 28, 47, 69, 97, 128, 165, 205, 251, 300, 355, ...
7, 24, 47, 80, 119, 170, 225, 292, 365, 448, 537, 638, ...
9, 35, 69, 119, 178, 257, 340, 443, 555, 683, 819, 975, ...
11, 48, 97, 170, 257, 372, 493, 644, 809, 998, 1197, 1426, ...
13, 63, 128, 225, 340, 493, 654, 857, 1078, 1331, 1595, 1901, ...
15, 80, 165, 292, 443, 644, 857, 1124, 1415, 1748, 2095, 2498, ...
17, 99, 205, 365, 555, 809, 1078, 1415, 1782, 2203, 2640, 3149, ...
19, 120, 251, 448, 683, 998, 1331, 1748, 2203, 2724, 3265, 3896, ...
21, 143, 300, 537, 819, 1197, 1595, 2095, 2640, 3265, 3914, 4673, ...
...
The initial antidiagonals are:
1
3, 3
5, 8, 5
7, 15, 15, 7
9, 24, 28, 24, 9
11, 35, 47, 47, 35, 11
13, 48, 69, 80, 69, 48, 13
15, 63, 97, 119, 119, 97, 63, 15
17, 80, 128, 170, 178, 170, 128, 80, 17
19, 99, 165, 225, 257, 257, 225, 165, 99, 19
21, 120, 205, 292, 340, 372, 340, 292, 205, 120, 21
23, 143, 251, 365, 443, 493, 493, 443, 365, 251, 143, 23
...

M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090.
M. Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5.
S. Legendre, The Number of Crossings in a Regular Drawing of the Complete Bipartite Graph, J. Integer Seqs., Vol. 12, 2009.
Index entries for sequences related to stained glass windows

This is one of a set of five arrays: A335678, A335679, A335680, A335681, A335682.
For the diagonal case see A306302 and A331757.
Cf. A114999, A331762.

Comment from Max Alekseyev, Jun 28 2020 (Start):
T(m,n) = 2*A114999(m-1,n-1) - A331762(m-1,n-1) + m*n + m + n - 2 for all m, n >= 1.  This follows from the Alekseyev-Basova-Zolotykh (2015) article.
Proof: Here is the appropriate modification of the corresponding comment in A306302: Assuming that K(m,n) has vertices at (i,0) and (j,1), for i=0..m-1 and j=0..n-1, the projective map (x,y) -> ((1-y)/(x+1), y/(x+1)) maps K(m,n) to the partition of the right triangle described by Alekseyev et al. (2015), for which Theorem 13 gives the number of regions, line segments, and intersection points. (End)

A335680 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of vertices in figure formed by taking m equally spaced points on a line and n equally spaced points on a parallel line, and joining each of the m points to each of the n points by a line segment.

Original entry on oeis.org

2, 3, 3, 4, 5, 4, 5, 8, 8, 5, 6, 12, 13, 12, 6, 7, 17, 21, 21, 17, 7, 8, 23, 30, 35, 30, 23, 8, 9, 30, 42, 51, 51, 42, 30, 9, 10, 38, 55, 73, 75, 73, 55, 38, 10, 11, 47, 71, 96, 109, 109, 96, 71, 47, 11, 12, 57, 88, 125, 143, 159, 143, 125, 88, 57, 12, 13, 68, 108, 156, 187, 209, 209, 187, 156, 108, 68, 13

Offset: 1

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Jun 28 2020

The case m=n (the main diagonal) is dealt with in A331755. A306302 has illustrations for the diagonal case for m = 1 to 15.

Also A335678 has colored illustrations for many values of m and n.

			The initial rows of the array are:
  2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ...
  3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80, ...
  4, 8, 13, 21, 30, 42, 55, 71, 88, 108, 129, 153, ...
  5, 12, 21, 35, 51, 73, 96, 125, 156, 192, 230, 274, ...
  6, 17, 30, 51, 75, 109, 143, 187, 234, 289, 346, 413, ...
  7, 23, 42, 73, 109, 159, 209, 274, 344, 426, 510, 609, ...
  8, 30, 55, 96, 143, 209, 275, 362, 455, 564, 674, 805, ...
  9, 38, 71, 125, 187, 274, 362, 477, 600, 744, 889, 1062, ...
  10, 47, 88, 156, 234, 344, 455, 600, 755, 937, 1119, 1337, ...
  11, 57, 108, 192, 289, 426, 564, 744, 937, 1163, 1389, 1660, ...
  12, 68, 129, 230, 346, 510, 674, 889, 1119, 1389, 1659, 1984, ...
  ...
The initial antidiagonals are:
  2
  3, 3
  4, 5, 4
  5, 8, 8, 5
  6, 12, 13, 12, 6
  7, 17, 21, 21, 17, 7
  8, 23, 30, 35, 30, 23, 8
  9, 30, 42, 51, 51, 42, 30, 9
  10, 38, 55, 73, 75, 73, 55, 38, 10
  11, 47, 71, 96, 109, 109, 96, 71, 47, 11
  12, 57, 88, 125, 143, 159, 143, 125, 88, 57, 12
  ...

M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090.
M. Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5.
S. Legendre, The Number of Crossings in a Regular Drawing of the Complete Bipartite Graph, J. Integer Seqs., Vol. 12, 2009.
Index entries for sequences related to stained glass windows

This is one of a set of five arrays: A335678, A335679, A335680, A335681, A335682.
For the diagonal case see A306302 and A331755.
Cf. A114999, A331762.

Comment from Max Alekseyev, Jun 28 2020 (Start):
T(m,n) = A114999(m-1,n-1) - A331762(m-1,n-1) + m + n for all m, n >= 1. This follows from the Alekseyev-Basova-Zolotykh (2015) article.
Proof: Here is the appropriate modification of the corresponding comment in A306302: Assuming that K(m,n) has vertices at (i,0) and (j,1), for i=0..m-1 and j=0..n-1, the projective map (x,y) -> ((1-y)/(x+1), y/(x+1)) maps K(m,n) to the partition of the right triangle described by Alekseyev et al. (2015), for which Theorem 13 gives the number of regions, line segments, and intersection points. (End)
Max Alekseyev's formula is an analog of Proposition 9 of Legendre (2009), and gives an explicit formula for this array. - N. J. A. Sloane, Jun 30 2020

A335681 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of interior vertices in figure formed by taking m equally spaced points on a line and n equally spaced points on a parallel line, and joining each of the m points to each of the n points by a line segment.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 6, 7, 6, 0, 0, 10, 14, 14, 10, 0, 0, 15, 22, 27, 22, 15, 0, 0, 21, 33, 42, 42, 33, 21, 0, 0, 28, 45, 63, 65, 63, 45, 28, 0, 0, 36, 60, 85, 98, 98, 85, 60, 36, 0, 0, 45, 76, 113, 131, 147, 131, 113, 76, 45, 0, 0, 55, 95, 143, 174, 196, 196, 174, 143, 95, 55, 0

Offset: 1

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Jun 28 2020

The case m=n (the main diagonal) is dealt with in A331755. A306302 has illustrations for the diagonal case for m = 1 to 15.

Also A335678 has colored illustrations for many values of m and n.

			The initial rows of the array are:
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
  0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ...
  0, 3, 7, 14, 22, 33, 45, 60, 76, 95, 115, 138, ...
  0, 6, 14, 27, 42, 63, 85, 113, 143, 178, 215, 258, ...
  0, 10, 22, 42, 65, 98, 131, 174, 220, 274, 330, 396, ...
  0, 15, 33, 63, 98, 147, 196, 260, 329, 410, 493, 591, ...
  0, 21, 45, 85, 131, 196, 261, 347, 439, 547, 656, 786, ...
  0, 28, 60, 113, 174, 260, 347, 461, 583, 726, 870, 1042, ...
  0, 36, 76, 143, 220, 329, 439, 583, 737, 918, 1099, 1316, ...
  0, 45, 95, 178, 274, 410, 547, 726, 918, 1143, 1368, 1638, ...
  0, 55, 115, 215, 330, 493, 656, 870, 1099, 1368, 1637, 1961, ...
  ...
The initial antidiagonals are:
  0
  0, 0
  0, 1, 0
  0, 3, 3, 0
  0, 6, 7, 6, 0
  0, 10, 14, 14, 10, 0
  0, 15, 22, 27, 22, 15, 0
  0, 21, 33, 42, 42, 33, 21, 0
  0, 28, 45, 63, 65, 63, 45, 28, 0
  0, 36, 60, 85, 98, 98, 85, 60, 36, 0
  0, 45, 76, 113, 131, 147, 131, 113, 76, 45, 0
  0, 55, 95, 143, 174, 196, 196, 174, 143, 95, 55, 0
  ...

Index entries for sequences related to stained glass windows

This is one of a set of five arrays: A335678, A335679, A335680, A335681, A335682.

For the diagonal case see A306302 and A331755.

It follows from the definitions that T(m,n) = A335680(m,n) - m - n. Note that there is an explicit formula for the latter sequence.

A335682 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of simple interior vertices in figure formed by taking m equally spaced points on a line and n equally spaced points on a parallel line, and joining each of the m points to each of the n points by a line segment.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 6, 6, 6, 0, 0, 10, 12, 12, 10, 0, 0, 15, 18, 24, 18, 15, 0, 0, 21, 27, 36, 36, 27, 21, 0, 0, 28, 36, 54, 54, 54, 36, 28, 0, 0, 36, 48, 72, 82, 82, 72, 48, 36, 0, 0, 45, 60, 96, 108, 124, 108, 96, 60, 45, 0, 0, 55, 75, 120, 144, 163, 163, 144, 120, 75, 55, 0

Offset: 1

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Jun 28 2020

A simple interior vertex is a vertex where exactly two lines cross.  In graph theory terms, this is an interior vertex of degree 4.
The case m=n (the main diagonal) is dealt with in A334701. A306302 has illustrations for the diagonal case for m = 1 to 15.
Also A335678 has colored illustrations for many values of m and n.
This is the only one of the five arrays (A335678-A335682) that does not have an explicit formula.
Let G_m(x) = g.f. for row m. For m <= 9, G_m appears to be a rational function of x with denominator D_m(x), where (writing C_k for the k-th cyclotomic polynomial):
D_3 = D_4 = C_1^3*C_2
D_5 = C_1^3*C_2*C_4
D_6 = C_1^3*C_2*C_4*C_5
D_7 = C_1^3*C_2*C_3*C_4*C_5*C_6
D_8 = D_9 = C_1^3*C_2*C_3*C_4*C_5*C_6*C_7

			The initial rows of the array are:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ...
0, 3, 6, 12, 18, 27, 36, 48, 60, 75, 90, 108, ...
0, 6, 12, 24, 36, 54, 72, 96, 120, 150, 180, 216, ...
0, 10, 18, 36, 54, 82, 108, 144, 180, 226, 270, 324, ...
0, 15, 27, 54, 82, 124, 163, 217, 272, 342, 408, 489, ...
0, 21, 36, 72, 108, 163, 214, 286, 358, 451, 536, 642, ...
0, 28, 48, 96, 144, 217, 286, 382, 478, 602, 715, 856, ...
0, 36, 60, 120, 180, 272, 358, 478, 598, 754, 894, 1070, ...
0, 45, 75, 150, 226, 342, 451, 602, 754, 950, 1126, 1347, ...
0, 55, 90, 180, 270, 408, 536, 715, 894, 1126, 1334, 1597, ...
0, 66, 108, 216, 324, 489, 642, 856, 1070, 1347, 1597, 1912, ...
...
The initial antidiagonals are:
0
0, 0
0, 1, 0
0, 3, 3, 0
0, 6, 6, 6, 0
0, 10, 12, 12, 10, 0
0, 15, 18, 24, 18, 15, 0
0, 21, 27, 36, 36, 27, 21, 0
0, 28, 36, 54, 54, 54, 36, 28, 0
0, 36, 48, 72, 82, 82, 72, 48, 36, 0
0, 45, 60, 96, 108, 124, 108, 96, 60, 45, 0
0, 55, 75, 120, 144, 163, 163, 144, 120, 75, 55, 0
...

Lars Blomberg, Table of n, a(n) for n = 1..9870 (the first 140 antidiagonals)
Index entries for sequences related to stained glass windows

This is one of a set of five arrays: A335678, A335679, A335680, A335681, A335682.

For the diagonal case see A306302, A331755, A334701.

A331771 a(n) = Sum_{-n

Original entry on oeis.org

0, 12, 56, 172, 400, 836, 1496, 2564, 4080, 6212, 8984, 12788, 17488, 23644, 31112, 40148, 50912, 64172, 79448, 97868, 118912, 143108, 170504, 202500, 238080, 278700, 323864, 374508, 430272, 493380, 561832, 638692, 722656, 814604, 914360, 1023428

Offset: 1

N. J. A. Sloane, Feb 08 2020

nonn

a(n) = 8*A332612(n)+4*n*(n-1)+4*(n-1)^2. Also adding 2 to the terms of the present sequence gives (essentially) A114146. - N. J. A. Sloane, Mar 14 2020

Koplowitz, Jack, Michael Lindenbaum, and A. Bruckstein. "The number of digital straight lines on an N* N grid." IEEE Transactions on Information Theory 36.1 (1990): 192-197. (See I(n).)

Seiichi Manyama, Table of n, a(n) for n = 1..1000
M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. See p. 158.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

When divided by 4 this becomes A115005, so this is a ninth sequence to add to the following list.
The following eight sequences are all essentially the same. The simplest is A115004(n), which we denote by z(n). Then A088658(n) = 4*z(n-1); A114043(n) = 2*z(n-1)+2*n^2-2*n+1; A114146(n) = 2*A114043(n); A115005(n) = z(n-1)+n*(n-1); A141255(n) = 2*z(n-1)+2*n*(n-1); A290131(n) = z(n-1)+(n-1)^2; A306302(n) = z(n)+n^2+2*n.
Cf. A332612.

VR := proc(m,n,q) local a,i,j; a:=0;
for i from -m+1 to m-1 do for j from -n+1 to n-1 do
if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
[seq(VR(n,n,1),n=1..50)];

a[n_] := Sum[Boole[GCD[i, j] == 1] (n - Abs[i]) (n - Abs[j]), {i, -n + 1, n - 1}, {j, -n + 1, n - 1}];
Array[a, 36] (* Jean-François Alcover, Apr 19 2020 *)

from sympy import totient
def A331771(n): return 4*((n-1)*(2*n-1)+sum(totient(i)*(n-i)*(2*n-i) for i in range(2,n))) # Chai Wah Wu, Aug 17 2021

a(n) = 4 * A115005(n).

a(n) = 4*((n-1)*(2n-1)+Sum_{i=2..n-1} (n-i)*(2*n-i)*phi(i)). - Chai Wah Wu, Aug 17 2021

A333277 a(n) = Sum_k k*A333275(n,k).

Original entry on oeis.org

4, 30, 116, 290, 652, 1186, 2092, 3370, 5212, 7546, 10828, 14866, 20260, 26674, 34508, 43778, 55364, 68546, 84644, 102962, 124172, 147842, 175772, 206738, 242372, 281506, 325652, 374090, 429356, 488938, 556348, 629738, 710468, 797210, 892492, 994514, 1107668

Offset: 1

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

nonn

a(n)/A331755(n) is the average number of polygons touching a non-boundary vertex in the graph defined in A306302.

Lars Blomberg, Table of n, a(n) for n = 1..100

Cf. A159065, A306302, A333274, A333275, A333276.

a(6) and beyond from Lars Blomberg, Jun 17 2020

A334694 a(n) = (n/4)(n^3+2n^2+5*n+8).

Original entry on oeis.org

0, 4, 17, 51, 124, 260, 489, 847, 1376, 2124, 3145, 4499, 6252, 8476, 11249, 14655, 18784, 23732, 29601, 36499, 44540, 53844, 64537, 76751, 90624, 106300, 123929, 143667, 165676, 190124, 217185, 247039, 279872, 315876, 355249, 398195, 444924, 495652, 550601, 609999, 674080, 743084, 817257, 896851, 982124, 1073340

Offset: 0

Scott R. Shannon and N. J. A. Sloane, May 19 2020

Consider a figure made up of a row of n >= 1 adjacent congruent rectangles in which all possible diagonals of the rectangles have been drawn. The number of regions formed is A306302. If we distort all these diagonals very slightly so that no three lines meet at a point, the number of regions changes to a(n).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for sequences related to stained glass windows
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A306302, A331452, A333275.

LinearRecurrence[{5,-10,10,-5,1},{0,4,17,51,124},50] (* or *)
A334694[n_]:=n/4(n^3+2n^2+5n+8);Array[A334694,50,0] (* Paolo Xausa, Nov 08 2023 *)

concat(0, Vec(x*(4 - 3*x + 6*x^2 - x^3) / (1 - x)^5 + O(x^40))) \\ Colin Barker, May 27 2020

Satisfies the identity a(n) = A306302(n) + Sum_{k=3..(n+1)} binomial(k-1,2)*A333275(n,2*k). E.g. for n=4 we have a(4) = 104 + 8*1 + 2*3 + 1*6 = 124.
From Colin Barker, May 27 2020: (Start)
G.f.: x*(4 - 3*x + 6*x^2 - x^3) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4.
(End)

cp's OEIS Frontend

A333281 Column 2 of triangle in A288180.

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A333282 Triangle read by rows: T(m,n) (m >= n >= 1) = number of regions formed by drawing the line segments connecting any two of the (m+1) X (n+1) lattice points in an m X n grid of squares and extending them to the boundary of the grid.

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A333276 a(n) = Sum_k k*A333274(n,k).

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A335679 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of edges in figure formed by taking m equally spaced points on a line and n equally spaced points on a parallel line, and joining each of the m points to each of the n points by a line segment.

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A335680 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of vertices in figure formed by taking m equally spaced points on a line and n equally spaced points on a parallel line, and joining each of the m points to each of the n points by a line segment.

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A335681 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of interior vertices in figure formed by taking m equally spaced points on a line and n equally spaced points on a parallel line, and joining each of the m points to each of the n points by a line segment.

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A335682 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of simple interior vertices in figure formed by taking m equally spaced points on a line and n equally spaced points on a parallel line, and joining each of the m points to each of the n points by a line segment.

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A331771 a(n) = Sum_{-n

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A333277 a(n) = Sum_k k*A333275(n,k).

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A334694 a(n) = (n/4)(n^3+2n^2+5*n+8).