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

4, 56, 208, 496, 1016, 1784, 2984, 4656, 6968, 9944, 13976, 18928, 25360, 33128, 42488, 53600, 67232, 82904, 101744, 123232, 147896, 175784, 208296, 244416, 285600, 331352, 382608, 439008, 502776, 571912, 649480, 734176, 826880, 927416, 1037288, 1155152, 1284992

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, Feb 08 2020

Keywords

• nonn

Comments

A "frame" of size n X n is formed from a grid of (n+1) X (n+1) points with the central grid of (n-3) X (n-3) points removed. If n is less than 3 then no points are removed, and a(n) = A255011(n). From now on we assume n >= 3.

If we focus on the squares rather than the points, the frame consists of an n X n array of squares with the central block of (n-2) X (n-2) squares removed.

The resulting structure has an outer perimeter with 4*n points and an inner perimeter with 4*n-8 points, for a total of 8*n-8 perimeter points. The frame itself is the strip of width 1 between the inner and outer perimeters.

Now join every pair of perimeter points, both inner and outer, by a line segment, provided the line remains inside the frame. The sequence gives the number of regions in the resulting figure.

Theorem. Let z(n) = Sum_{i, j = 1..n, gcd(i,j)=1} (n+1-i)*(n+1-j) (this is A115004). Then, for n >= 2, a(n) = 4*z(n) + 16*n^2 - 20*n. - 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(1) = 4
• Scott R. Shannon, Colored illustration for a(2) = 56
• Scott R. Shannon, Colored illustration for a(3) = 208
• Scott R. Shannon, Colored illustration for a(4) = 496
• Scott R. Shannon, Colored illustration for a(5) = 1016
• Scott R. Shannon, Colored illustration for a(6) = 1784
• Scott R. Shannon, Colored illustration for a(7) = 2984
• Scott R. Shannon, Colored illustration for a(8) = 4656
• Scott R. Shannon, Colored illustration for a(8) = 4656 (Another version)
• Zach Shannon, Illustration for a(8) = 4656 used as a frame for the OEIS logo
• Zach Shannon, Illustration for a(8) = 4656 used as a frame for the OEIS logo (detail)
• N. J. A. Sloane, Illustration for a(3) = 208

Crossrefs

This is the main diagonal of A331457. Equals 4 times A332594.

Cf. A255011, A115004.

The analogous sequence for an n X n block of squares (if the center block is not removed) is A331452.

Programs

Maple

# First define z(n) = A115004
z := proc(n)
    local a, b, r ;
    r := 0 ;
    for a from 1 to n do
    for b from 1 to n do
        if igcd(a, b) = 1 then
            r := r+(n+1-a)*(n+1-b);
        end if;
    end do:
    end do:
    r ;
end proc:
A331776 := n -> if n=1 then 4 else 4*z(n)+16*n^2 - 20*n; fi;
[seq(A331776(n),n=1..40)]; # N. J. A. Sloane, Mar 09 2020

PARI

a(n) = 4*sum(i=1, n, sum(j=1, n, if(gcd(i, j)==1, (n+1-i)*(n+1-j), 0))) + 16*n^2 - 20*n + 4*(n==1); \\ Jinyuan Wang, Aug 07 2021

Python

from sympy import totient
def A331776(n): return 4 if n == 1 else 20*n*(n-1) + 4*sum(totient(i)*(n+1-i)*(2*n+2-i) for i in range(2,n+1)) # Chai Wah Wu, Aug 16 2021

Formula

For n > 1, a(n) = 20*n*(n-1) + 4*Sum_{i=2..n} (n+1-i)*(2n+2-i)*phi(i). - Chai Wah Wu, Aug 16 2021

Extensions

More terms from N. J. A. Sloane, Mar 09 2020