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A332598
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Number of vertices in a "frame" of size n X n (see Comments in A331776 for definition).
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3
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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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
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MAPLE
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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;
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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