A334694 -id:A334694 - cp's OEIS frontend

A306302 Number of regions into which a figure made up of a row of n adjacent congruent rectangles is divided upon drawing diagonals of all possible rectangles (a(0)=0 by convention).

0, 4, 16, 46, 104, 214, 380, 648, 1028, 1562, 2256, 3208, 4384, 5924, 7792, 10052, 12744, 16060, 19880, 24486, 29748, 35798, 42648, 50648, 59544, 69700, 80992, 93654, 107596, 123374, 140488, 159704, 180696, 203684, 228624, 255892, 285152, 317400, 352096, 389576

Offset: 0

Paarth Jain, Feb 05 2019

nonn

Assuming that the rectangles have vertices at (k,0) and (k,1), k=0..n, the projective map (x,y) -> ((1-y)/(x+1),y/(x+1)) maps their partition to the partition of the right isosceles triangle described by Alekseyev et al. (2015), for which Theorem 13 gives the number of regions, line segments, and intersection points. - Max Alekseyev, Apr 10 2019

The figure is made up of A324042 triangles and A324043 quadrilaterals. - N. J. A. Sloane, Mar 03 2020

Jinyuan Wang, Table of n, a(n) for n = 0..1000
Max Alekseyev, Illustration for n = 3.
M. A. Alekseyev. On the number of two-dimensional threshold functions, arXiv:math/0602511 [math.CO], 2006-2010; doi:10.1137/090750184, SIAM J. Disc. Math. 24(4), 2010, pp. 1617-1631.
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.
Lars Blomberg, Scott R. Shannon and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.
M. Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5, Lemma 2.
Robert Israel, Maple program, Feb 07 2019
S. Legendre, The Number of Crossings in a Regular Drawing of the Complete Bipartite Graph, J. Integer Seqs., Vol. 12, 2009.
Scott R. Shannon, Colored illustration for T(1,1)
Scott R. Shannon, Colored illustration for T(2,1)
Scott R. Shannon, Colored illustration for T(3,1)
Scott R. Shannon, Colored illustration for T(4,1)
Scott R. Shannon, Colored illustration for T(5,1)
Scott R. Shannon, Colored illustration for T(6,1)
Scott R. Shannon, Colored illustration for T(7,1)
Scott R. Shannon, Colored illustration for T(8,1)
Scott R. Shannon, Colored illustration for T(9,1)
Scott R. Shannon, Colored illustration for T(10,1)
Scott R. Shannon, Colored illustration for T(11,1)
Scott R. Shannon, Colored illustration for T(12,1)
Scott R. Shannon, Colored illustration for T(13,1)
Scott R. Shannon, Colored illustration for T(14,1)
Scott R. Shannon, Colored illustration for T(15,1)
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Index entries for sequences related to stained glass windows

See A331755 for the number of vertices, A331757 for the number of edges.
Cf. A007678, A108914, A114043, A324042, A324043, A333286, A333287, A333288, A334694.
A column of A288187. See A288177 for additional references.
Also a column of A331452 and A356790.
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. - N. J. A. Sloane, Feb 04 2020

# Maple from N. J. A. Sloane, Mar 04 2020, starting at n=1:  First define z(n) = A115004
z := proc(n)
    local a, b, r ;
    r := 0 ;
    for a from 1 to n do
    for b from 1 to n do
        if igcd(a, b) = 1 then
            r := r+(n+1-a)*(n+1-b);
        end if;
    end do:
    end do:
    r ;
end proc:
a := n-> z(n)+n^2+2*n;
[seq(a(n), n=1..50)];

z[n_] := Sum[(n - i + 1)(n - j + 1) Boole[GCD[i, j] == 1], {i, n}, {j, n}];
a[0] = 0;
a[n_] := z[n] + n^2 + 2n;
a /@ Range[0, 40] (* Jean-François Alcover, Mar 24 2020 *)

from sympy import totient
def A306302(n): return 2*n*(n+1) + sum(totient(i)*(n+1-i)*(2*n+2-i) for i in range(2,n+1)) # Chai Wah Wu, Aug 16 2021

a(n) = n + (A114043(n+1) - 1)/2, conjectured by N. J. A. Sloane, Feb 07 2019; proved by Max Alekseyev, Apr 10 2019
a(n) = n + A115005(n+1) = n + A141255(n+1)/2. - Max Alekseyev, Apr 10 2019
a(n) = A324042(n) + A324043(n). - Jinyuan Wang, Mar 19 2020
a(n) = Sum_{i=1..n, j=1..n, gcd(i,j)=1} (n+1-i)*(n+1-j) + n^2 + 2*n. - N. J. A. Sloane, Apr 11 2020
a(n) = 2n(n+1) + Sum_{i=2..n} (n+1-i)*(2n+2-i)*phi(i). - Chai Wah Wu, Aug 16 2021

a(6)-a(20) from Robert Israel, Feb 07 2019
Edited and more terms added by Max Alekseyev, Apr 10 2019
a(0) added by N. J. A. Sloane, Feb 04 2020

A333275 Irregular triangle read by rows: consider the graph defined in A306302 formed from a row of n adjacent congruent rectangles by drawing the diagonals of all visible rectangles; T(n,k) (n >= 1, 2 <= k <= 2n+2) is the number of non-boundary vertices in the graph at which k polygons meet.

Original entry on oeis.org

0, 0, 1, 0, 0, 6, 0, 1, 0, 0, 24, 0, 2, 0, 1, 0, 0, 54, 0, 8, 0, 2, 0, 1, 0, 0, 124, 0, 18, 0, 2, 0, 2, 0, 1, 0, 0, 214, 0, 32, 0, 10, 0, 2, 0, 2, 0, 1, 0, 0, 382, 0, 50, 0, 22, 0, 2, 0, 2, 0, 2, 0, 1, 0, 0, 598, 0, 102, 0, 18, 0, 12, 0, 2, 0, 2, 0, 2, 0, 1

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 14 2020

The number of polygons meeting at a non-boundary vertex is simply the degree (or valency) of that vertex.
Row sums are A159065.
Sum_k k*T(n,k) gives A333277.
See A333274 for the degrees if the boundary vertices are included.
T(n,k) = 0 if k is odd. But the triangle includes those zero entries because this is used to construct A333274.

			Led d denote the number of polygons meeting at a vertex.
For n=2, in the interiors of each of the two squares there are 3 points with d=4, and the center point has d=6.
So in total there are 6 points with d=4 and 1 with d=6. So row 2 of the triangle is [0, 0, 6, 0, 1].
The triangle begins:
0,0,1,
0,0,6,0,1,
0,0,24,0,2,0,1,
0,0,54,0,8,0,2,0,1,
0,0,124,0,18,0,2,0,2,0,1,
0,0,214,0,32,0,10,0,2,0,2,0,1,
0,0,382,0,50,0,22,0,2,0,2,0,2,0,1,
0,0,598,0,102,0,18,0,12,0,2,0,2,0,2,0,1
...
If we leave out the uninteresting zeros, the triangle begins:
[1]
[6, 1]
[24, 2, 1]
[54, 8, 2, 1]
[124, 18, 2, 2, 1]
[214, 32, 10, 2, 2, 1]
[382, 50, 22, 2, 2, 2, 1]
[598, 102, 18, 12, 2, 2, 2, 1]
[950, 126, 32, 26, 2, 2, 2, 2, 1]
[1334, 198, 62, 20, 14, 2, 2, 2, 2, 1]
[1912, 286, 100, 10, 30, 2, 2, 2, 2, 2, 1]
[2622, 390, 118, 38, 22, 16, 2, 2, 2, 2, 2, 1]
... - _N. J. A. Sloane_, Jul 27 2020

Lars Blomberg, Table of n, a(n) for n = 1..10200 (the first 100 rows)
Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.
Scott R. Shannon, Colored illustration showing regions for n=1
Scott R. Shannon, Images of vertices for n=1.
Scott R. Shannon, Colored illustration showing regions for n=2
Scott R. Shannon, Images of vertices for n=2.
Scott R. Shannon, Colored illustration showing regions for n=3
Scott R. Shannon, Images of vertices for n=3.
Scott R. Shannon, Colored illustration showing regions for n=4
Scott R. Shannon, Images of vertices for n=4.
Scott R. Shannon, Colored illustration showing regions for n=5
Scott R. Shannon, Images of vertices for n=5
Scott R. Shannon, Colored illustration showing regions for n=6
Scott R. Shannon, Images of vertices for n=6
Scott R. Shannon, Images of vertices for n=7
Scott R. Shannon, Images of vertices for n=8
Scott R. Shannon, Images of vertices for n=9.
Scott R. Shannon, Images of vertices for n=11.
Scott R. Shannon, Images of vertices for n=14.
Index entries for sequences related to stained glass windows

Cf. A306302, A331755, A331757, A331452, A333274, A333276, A333277, A334694.

a(36) and beyond from Lars Blomberg, Jun 17 2020

A334691 Irregular triangle read by rows: T(n,k) (n >= 1, 2 <= k <= 2*n) = number of interior vertices in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter) where k lines meet.

Original entry on oeis.org

1, 20, 8, 1, 204, 32, 8, 0, 1, 616, 152, 20, 8, 4, 0, 1, 2428, 252, 36, 16, 4, 0, 0, 0, 1, 3968, 572, 156, 72, 16, 8, 4, 0, 4, 0, 1, 11164, 900, 120, 52, 16, 8, 4, 0, 0, 0, 0, 0, 1, 16884, 1712, 396, 132, 40, 20, 8, 8, 8, 0, 0, 0, 0, 0, 1, 30116, 2536, 600, 140, 60, 24, 8, 20, 8, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 18 2020

No formula is known.

			Triangle begins:
1;
20,8,1;
204,32,8,0,1;
616,152,20,8,4,0,1;
2428,252,36,16,4,0,0,0,1;
3968,572,156,72,16,8,4,0,4,0,1;
11164,900,120,52,16,8,4,0,0,0,0,0,1;
16884,1712,396,132,40,20,8,8,8,0,0,0,0,0,1;
30116,2536,600,140,60,24,8,20,8,0,0,0,0,0,0,0,1;
43988,4056,948,312,84,56,52,20,,8,0,0,4,0,0,0,0,0,1;
82016,4660,580,228,48,84,4,4,4,8,4,0,0,0,0,0,0,0,0,0,1;
90088,8504,1840,780,424,128,68,32,32,0,0,8,24,0,0,0,4,0,0,0,0,0,1;
168360,8284,1056,396,128,100,52,12,4,4,4,8,4,0,0,0,0,0,0,0,0,0,0,0,1;
202332,13144,2980,924,256,144,140,60,44,4,0,8,8,8,0,0,4,0,0,0,0,0,0,0,0,0,1;
...

Scott R. Shannon, Data for triangles A334691 and A334699
Scott R. Shannon, Colored illustration for n = 2
Scott R. Shannon, Illustration for n=3 showing interior vertices color-coded according to multiplicity.

Cf. A255011, A331449, A334690 (row sums), A334692 (column k=2), A334693 (k=3), A334694-A334699.

cp's OEIS Frontend

A306302 Number of regions into which a figure made up of a row of n adjacent congruent rectangles is divided upon drawing diagonals of all possible rectangles (a(0)=0 by convention).

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A333275 Irregular triangle read by rows: consider the graph defined in A306302 formed from a row of n adjacent congruent rectangles by drawing the diagonals of all visible rectangles; T(n,k) (n >= 1, 2 <= k <= 2n+2) is the number of non-boundary vertices in the graph at which k polygons meet.

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Extensions

A334691 Irregular triangle read by rows: T(n,k) (n >= 1, 2 <= k <= 2*n) = number of interior vertices in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter) where k lines meet.

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