A088658 Number of triangles in an n X n unit grid that have minimal possible area (of 1/2).

0, 4, 32, 124, 320, 716, 1328, 2340, 3792, 5852, 8544, 12260, 16864, 22916, 30272, 39188, 49824, 62948, 78080, 96348, 117232, 141260, 168480, 200292, 235680, 276100, 321056, 371484, 427024, 489900, 558112, 634724, 718432, 810116, 909600, 1018388, 1135136, 1263828, 1402304, 1551908

Offset

1,2

Links

Ray Chandler, Table of n, a(n) for n = 1..1000

N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)

Formula

a(n+1) = 4*A115004(n).

a(n) = 4*(n-1)^2 + 4*Sum_{i=2..n-1} (n-i)*(2n-i)*phi(i). - Chai Wah Wu, Aug 15 2021

Example

a(2)=4 because 4 (isosceles right) triangles with area 1/2 can be placed on a 2 X 2 grid.

Mathematica

z[n_] := Sum[(n - i + 1)(n - j + 1) Boole[GCD[i, j] == 1], {i, n}, {j, n}];
a[n_] := 4 z[n - 1];
Array[a, 40] (* Jean-François Alcover, Mar 24 2020 *)

Prog

(Python)
from sympy import totient
def A088658(n): return 4*(n-1)**2 + 4*sum(totient(i)*(n-i)*(2*n-i) for i in range(2, n)) # Chai Wah Wu, Aug 15 2021

Crossrefs

Cf. A045996.

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: A267668 A338322 A239056 * A088802 A123854 A301843

Adjacent sequences: A088655 A088656 A088657 * A088659 A088660 A088661

Keyword

nonn

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 21 2003

Extensions

a(7)-a(28) from Ray Chandler, May 03 2011

Corrected and extended by Ray Chandler, May 18 2011

Status

approved