A356790 -id:A356790 - 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

A355902 Start with a 2 X n array of squares, join every vertex on top edge to every vertex on bottom edge; a(n) = one-half the number of cells.

Original entry on oeis.org

0, 3, 10, 26, 56, 112, 196, 331, 522, 790, 1138, 1615, 2204, 2975, 3910, 5041, 6388, 8047, 9958, 12262, 14894, 17920, 21346, 25347, 29796, 34875, 40522, 46854, 53826, 61716, 70274, 79883, 90380, 101875, 114346, 127981, 142612, 158737, 176086, 194827, 214852, 236717, 259906, 285124, 311970, 340588, 370990, 403819, 438440, 475556

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Sep 05 2022

nonn

Note that this figure can be obtained by drawing an "equatorial" line through the middle of the strip of n adjacent rectangles in A306302. This cuts each of the 2n "equatorial" cells in A306302 in two. It follows that 2*a(n) = A306302(n) + 2*n, i.e. that a(n) = A306302(n)/2 + n. Note that there is an explicit formula for A306302(n) in terms of n. - Scott R. Shannon, Sep 06 2022.

This means the present sequence is one more member of the large class of sequences which are essentially the same as A115004 (see Cross-References). - N. J. A. Sloane, Sep 06 2022

Scott R. Shannon, Image for a(4) = 56. Note in this and other images the entire 2xn array is shown so the number of cells is twice a(n).
Scott R. Shannon, Image for a(6) = 196.
Scott R. Shannon, Image for a(10) = 1138.
Scott R. Shannon, Image for a(15) = 5041.
N. J. A. Sloane, Illustration for a(2) = 10.
N. J. A. Sloane, Illustration for a(3) = 26.

Cf. A356790, A306302, A355798, A290131, A331452.

The following nine 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; A355902(n) = n + A306302(n)/2. - N. J. A. Sloane, Sep 06 2022

a(n) = A356790(2,n+2)/2 - 2.

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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