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A274462
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Place n equally-spaced points around a circle, labeled 0,1,2,...,n-1. For each i = 0..n-1 such that 4i != i mod n, draw an (undirected) chord from i to (4i mod n). Then a(n) is the total number of distinct chords.
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3
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0, 0, 1, 0, 3, 2, 3, 6, 7, 6, 7, 10, 9, 12, 13, 6, 15, 16, 15, 18, 17, 18, 21, 22, 21, 22, 25, 24, 27, 28, 21, 30, 31, 30, 33, 32, 33, 36, 37, 36, 37, 40, 39, 42, 43, 36, 45, 46, 45, 48, 47, 48, 51, 52, 51, 52, 55, 54, 57, 58, 51
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OFFSET
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0,5
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LINKS
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FORMULA
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We argue as in A273724. There are n-1 choices for i.
For nontrivial chords we need i != 4i mod n, which means 3i != 0 mod n, and so when n == 0 mod 3 we must subtract 2 from n-1.
A chord occurs twice (but must be counted only once) when j==4i mod n and i==4j mod n, thus when 15i==0 mod n. If n==+/- 5 mod 15 then subtract another 2, if n==0 mod 15 subtract 6.
Putting the pieces together, we obtain the g.f.
8 + x^2/(1-x)^2 - 2/(1-x^3) - 2(x^5+x^10)/(1-x^15) - 6/(1-x^15),
which can be rewritten as
x^2*(9*x^14-7*x^13+x^12+3*x^11-x^10+3*x^9+x^8-x^7+x^6+3*x^5+x^4-x^3+3*x^2-x+1)/((1-x)*(1-x^15)).
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MAPLE
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M:=4; # M is the multiplier (2 for A117571, 3 for A273724, 4 for the present sequence)
ans:=[0, 0];
for n from 2 to 100 do
h:=Array(0..n-1, 0..n-1, 0); ct:=0;
for i from 1 to n-1 do j := (M*i mod n);
if i<j then if h[i, j]=0 then ct:=ct+1; h[i, j]:=1; fi;
elif i>j then if h[j, i]=0 then ct:=ct+1; h[j, i]:=1; fi;
fi;
od:
ans:=[op(ans), ct];
od:
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CROSSREFS
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If 4i in the definition is replaced by 2i we get A117571, and if 4i is replaced by 3i we get A273724.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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