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A331776
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Number of regions in a "frame" of size n X n (see Comments for definition).
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14
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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MAPLE
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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;
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PROG
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(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
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CROSSREFS
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The analogous sequence for an n X n block of squares (if the center block is not removed) is A331452.
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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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