A331757 Number of edges in a figure made up of a row of n adjacent congruent rectangles upon drawing diagonals of all possible rectangles.

8, 28, 80, 178, 372, 654, 1124, 1782, 2724, 3914, 5580, 7626, 10352, 13590, 17540, 22210, 28040, 34670, 42760, 51962, 62612, 74494, 88508, 104042, 121912, 141534, 163664, 187942, 215636, 245490, 279260, 316022, 356456, 399898, 447612, 498698, 555352

Offset: 1

Author

N. J. A. Sloane, Feb 04 2020

Keywords

• nonn

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• Lars Blomberg, Scott R. Shannon and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.

Crossrefs

A306302 gives number of regions in the figure.

This is column 1 of A331454.

Cf. A324042, A324043.

Programs

Mathematica

Table[n^2 + 4n + 1 + Sum[Sum[(2 * Boole[GCD[i, j] == 1] - Boole[GCD[i, j] == 2]) * (n + 1 - i) * (n + 1 - j), {j, 1, n}], {i, 1, n}], {n, 1, 37}] (* Joshua Oliver, Feb 05 2020 *)

Python

from sympy import totient
def A331757(n): return 8 if n == 1 else 2*(n*(n+3) + sum(totient(i)*(n+1-i)*(n+1+i) 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) = (2*n + 2 + 3*A324042(n) + 4*A324043(n))/2 [Corrected by Chai Wah Wu, Aug 16 2021]

For n > 1, a(n) = 2*(n*(n+3) + Sum_{i=2..floor(n/2)} (n+1-i)*(n+1+i)*phi(i) + Sum_{i=floor(n/2)+1..n} (n+1-i)*(2*n+2-i)*phi(i)). - Chai Wah Wu, Aug 16 2021