A324043 Number of quadrilateral regions into which a figure made up of a row of n adjacent congruent rectangles is divided upon drawing diagonals of all possible rectangles.
0, 2, 14, 34, 90, 154, 288, 462, 742, 1038, 1512, 2074, 2904, 3774, 4892, 6154, 7864, 9662, 12022, 14638, 17786, 20998, 25024, 29402, 34672, 40038, 46310, 53038, 61090, 69454, 79344, 89890, 101792, 113854, 127476, 141866, 158428, 175182, 193760, 213274, 235444, 258182, 283858, 310750, 339986
Offset: 1
Keywords
Examples
For k adjacent congruent rectangles, the number of quadrilateral regions in the j-th rectangle is: k\j| 1 2 3 4 5 6 7 ... ---+-------------------------------- 1 | 0, 0, 0, 0, 0, 0, 0, ... 2 | 1, 1, 0, 0, 0, 0, 0, ... 3 | 3, 8, 3, 0, 0, 0, 0, ... 4 | 5, 12, 12, 5, 0, 0, 0, ... 5 | 7, 22, 32, 22, 7, 0, 0, ... 6 | 9, 28, 40, 40, 28, 9, 0, ... 7 | 11, 38, 58, 74, 58, 38, 11, ... ... a(4) = 5 + 12 + 12 + 5 = 34.
Links
- Chai Wah Wu, Table of n, a(n) for n = 1..10000
- 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.
- Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.
- Robert Israel, Maple program.
- Jinyuan Wang, Illustration for n = 1, 2, 3, 4, 5.
Crossrefs
Programs
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Maple
See Robert Israel link. There are also Maple programs for both A306302 and A324042. Then a := n -> A306302(n) - A324042(n); # N. J. A. Sloane, Mar 04 2020
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Mathematica
Table[Sum[Sum[(Boole[GCD[i, j] == 1] - 2 * Boole[GCD[i, j] == 2]) * (n + 1 - i) * (n + 1 - j), {j, 1, n}], {i, 1, n}] - n^2, {n, 1, 45}] (* Joshua Oliver, Feb 05 2020 *) -
PARI
{ A324043(n) = sum(i=1, n, sum(j=1, n, ( (gcd(i, j)==1) - 2*(gcd(i,j)==2) ) * (n+1-i) * (n+1-j) )) - n^2; } \\ Max Alekseyev, Jul 08 2019 -
Python
from sympy import totient def A324043(n): return 0 if n==1 else -2*(n-1)**2 + sum(totient(i)*(n+1-i)*(7*i-2*n-2) for i in range(2,n//2+1)) + sum(totient(i)*(n+1-i)*(2*n+2-i) for i in range(n//2+1,n+1)) # Chai Wah Wu, Aug 16 2021
Formula
a(n) = A115005(n+1) - A177719(n+1) - n - 1 = Sum_{i,j=1..n; gcd(i,j)=1} (n+1-i)*(n+1-j) - 2*Sum_{i,j=1..n; gcd(i,j)=2} (n+1-i)*(n+1-j) - n^2. - Max Alekseyev, Jul 08 2019
For n>1, a(n) = -2(n-1)^2 + Sum_{i=2..floor(n/2)} (n+1-i)*(7i-2n-2)*phi(i) + Sum_{i=floor(n/2)+1..n} (n+1-i)*(2n+2-i)*phi(i). - Chai Wah Wu, Aug 16 2021
Extensions
a(8)-a(23) from Robert Israel, Jul 07 2019
Terms a(24) onward from Max Alekseyev, Jul 08 2019
Comments