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

1,1

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.

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

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 *)

Prog

(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

Crossrefs

A306302 gives number of regions in the figure.

This is column 1 of A331454.

Cf. A324042, A324043.

Sequence in context: A134638 A293289 A305638 * A130129 A317032 A229713

Adjacent sequences: A331754 A331755 A331756 * A331758 A331759 A331760

Keyword

nonn

Author

N. J. A. Sloane, Feb 04 2020

Status

approved