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A306302
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Number of regions into which a figure made up of a row of n adjacent congruent rectangles is divided upon drawing diagonals of all possible rectangles (a(0)=0 by convention).
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51
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0, 4, 16, 46, 104, 214, 380, 648, 1028, 1562, 2256, 3208, 4384, 5924, 7792, 10052, 12744, 16060, 19880, 24486, 29748, 35798, 42648, 50648, 59544, 69700, 80992, 93654, 107596, 123374, 140488, 159704, 180696, 203684, 228624, 255892, 285152, 317400, 352096, 389576
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OFFSET
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0,2
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COMMENTS
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Assuming that the rectangles have vertices at (k,0) and (k,1), k=0..n, the projective map (x,y) -> ((1-y)/(x+1),y/(x+1)) maps their partition to the partition of the right isosceles triangle described by Alekseyev et al. (2015), for which Theorem 13 gives the number of regions, line segments, and intersection points. - Max Alekseyev, Apr 10 2019
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LINKS
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FORMULA
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a(n) = Sum_{i=1..n, j=1..n, gcd(i,j)=1} (n+1-i)*(n+1-j) + n^2 + 2*n. - N. J. A. Sloane, Apr 11 2020
a(n) = 2n(n+1) + 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:
a := n-> z(n)+n^2+2*n;
[seq(a(n), n=1..50)];
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MATHEMATICA
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z[n_] := Sum[(n - i + 1)(n - j + 1) Boole[GCD[i, j] == 1], {i, n}, {j, n}];
a[0] = 0;
a[n_] := z[n] + n^2 + 2n;
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PROG
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(Python)
from sympy import totient
def A306302(n): return 2*n*(n+1) + 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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See A331755 for the number of vertices, A331757 for the number of edges.
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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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