A332598 Number of vertices in a "frame" of size n X n (see Comments in A331776 for definition).

5, 27, 152, 364, 776, 1340, 2272, 3532, 5336, 7516, 10592, 14316, 19328, 25100, 32176, 40428, 50848, 62476, 76824, 93020, 111880, 132492, 157056, 184140, 215552, 249452, 287928, 329900, 378216, 429852, 488768, 552572, 623104, 697884, 780464, 868588, 967056

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 >= 3, a(n) = 4*z(n) - 4*z_2(n) + 12*n^2 - 24*n + 8. (This does not hold for n<3, because it uses Euler's formula, and the graph for n<3 has no hole, so has genus 0, whereas for n>=3 there is a hole and the graph has genus 1.) - Scott R. Shannon and N. J. A. Sloane, Mar 04 2020

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..1000
• Scott R. Shannon, Colored illustration for a(3) = 152 (there are 152 vertices in this picture).

Crossrefs

Cf. A331776 (regions), A332597 (edges).

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 5 elif n=2 then 27 else 12*n^2 - 24*n + 8 + 4*V(n,n,1) - 4*V(n, n, 2); fi;
[seq(f(n), n=1..50)]; # N. J. A. Sloane, Mar 10 2020

PARI

a(n) = if(n<3, 22*n - 17, 4*sum(i=1, n, sum(j=1, n, if(gcd(i, j)==1, (n+1-i)*(n+1-j), 0))) - 4*sum(i=1, n, sum(j=1, n, if(gcd(i, j)==2, (n+1-i)*(n+1-j), 0))) + 12*n^2 - 24*n + 8); \\ Jinyuan Wang, Aug 07 2021

Python

from sympy import totient
def A332598(n): return 22*n-17 if n <= 2 else 4*(n-1)*(3*n-1) + 12*sum(totient(i)*(n+1-i)*i for i in range(2,n//2+1)) + 4*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 > 2, a(n) = 4*(n-1)*(3n-1)+12*Sum_{i=2..floor(n/2)} (n+1-i)*i*phi(i) + 4*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