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

1,1

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).

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

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

Prog

(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

Crossrefs

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

Sequence in context: A220573 A303743 A187157 * A331448 A215057 A222400

Adjacent sequences: A332594 A332595 A332596 * A332598 A332599 A332600

Keyword

nonn

Author

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

Extensions

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

Status

approved