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

0,2

Comments

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

Links

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

Formula

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

Maple

# 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)];

Mathematica

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 *)

Prog

(Python)
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

Crossrefs

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

Sequence in context: A213501 A097125 A213480 * A159940 A000704 A007315

Adjacent sequences: A306299 A306300 A306301 * A306303 A306304 A306305

Keyword

nonn

Author

Paarth Jain, Feb 05 2019

Extensions

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

Status

approved