A332597 Number of edges in a "frame" of size n X n (see Comments in A331776 for definition).

8, 92, 360, 860, 1792, 3124, 5256, 8188, 12304, 17460, 24568, 33244, 44688, 58228, 74664, 94028, 118080, 145380, 178568, 216252, 259776, 308276, 365352, 428556, 501152, 580804, 670536, 768908, 880992, 1001764, 1138248, 1286748, 1449984, 1625300, 1817752, 2023740, 2252048, 2495476, 2759304, 3040460, 3349056

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, Mar 02 2020

Keywords

• nonn

Comments

See A331776 for many other illustrations.

Theorem. Let z(n) = Sum_{i, j = 1..n, gcd(i,j)=1} (n+1-i)*(n+1-j) (this is A115004) and z_2(n) = Sum_{i, j = 1..n, gcd(i,j)=2} (n+1-i)*(n+1-j) (this is A331761). Then, for n >= 2, a(n) = 8*z(n) - 4*z_2(n) + 28*n^2 - 44*n + 8. - Scott R. Shannon and N. J. A. Sloane, Mar 06 2020

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• Scott R. Shannon, Colored illustration for a(3) = 360 (there are 360 edges in this picture).

Crossrefs

Cf. A115004, A331761, A331776 (regions), A332598 (vertices).

Programs

Maple

V := proc(m, n, q) local a, i, j; a:=0;
for i from 1 to m do for j from 1 to n do
if gcd(i, j)=q then a:=a+(m+1-i)*(n+1-j); fi; od: od: a; end;
f := n -> if n=1 then 8 else 28*n^2 - 44*n + 8 + 8*V(n,n,1) - 4*V(n, n, 2); fi;
[seq(f(n), n=1..50)]; # N. J. A. Sloane, Mar 10 2020

Python

from sympy import totient
def A332597(n): return 8 if n == 1 else 4*(n-1)*(8*n-1) + 8*sum(totient(i)*(n+1-i)*(n+i+1) for i in range(2,n//2+1)) + 8*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

For n > 1, a(n) = 4*(n-1)*(8*n-1) + 8*Sum_{i=2..floor(n/2)} (n+1-i)*(n+i+1)*phi(i) + 8*Sum_{i=floor(n/2)+1..n} (n+1-i)*(2*n+2-i)*phi(i). - Chai Wah Wu, Aug 16 2021

Extensions

More terms from N. J. A. Sloane, Mar 10 2020