A332596 Number of quadrilateral regions in a "frame" of size n X n (see Comments in A331776 for definition), divided by 8.

0, 1, 10, 26, 63, 107, 189, 294, 455, 627, 891, 1202, 1650, 2121, 2719, 3392, 4292, 5239, 6470, 7832, 9463, 11129, 13205, 15460, 18164, 20919, 24130, 27572, 31679, 35945, 40977, 46340, 52384, 58511, 65421, 72718, 81104, 89589, 98989, 108860, 120062, 131551

Offset

1,3

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, 8*a(n) = 4*z(n) - 8*z_2(n) + 8*n^2 - 36*n + 24. - Scott R. Shannon and N. J. A. Sloane, Mar 06 2020

Links

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Scott R. Shannon, Colored illustration for a(3) = 10 (there are 8*10 quadrilaterals).

Formula

For n > 1, a(n) = ((n-1)*(n-4) - Sum_{i=2..floor(n/2)} (n+1-i)*(2*n+2-7*i)*phi(i) + Sum_{i=floor(n/2)+1..n} (n+1-i)*(2*n+2-i)*phi(i))/2. - 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 0 else 8*n^2 - 36*n + 24 + 4*V(n, n, 1) 8*V(n, n, 2); fi;
[seq(f(n)/8, n=1..60)]; # N. J. A. Sloane, Mar 10 2020

Prog

(PARI) a(n) = sum(i=1, n, sum(j=1, n, if(gcd(i, j)==1, (n+1-i)*(n+1-j), 0)))/2 - sum(i=1, n, sum(j=1, n, if(gcd(i, j)==2, (n+1-i)*(n+1-j), 0))) + n^2 - 9*n/2 + 3; \\ Jinyuan Wang, Aug 07 2021
(Python)
from sympy import totient
def A332596(n): return 0 if n == 1 else ((n-1)*(n-4) - sum(totient(i)*(n+1-i)*(2*n+2-7*i) for i in range(2, n//2+1)) + sum(totient(i)*(n+1-i)*(2*n+2-i) for i in range(n//2+1, n+1)))//2 # Chai Wah Wu, Aug 16 2021

Crossrefs

Cf. A115004, A331761, A331776, A332594, A332595.

Sequence in context: A125075 A209983 A055710 * A134420 A144255 A259290

Adjacent sequences: A332593 A332594 A332595 * A332597 A332598 A332599

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