A274462 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.

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

Offset: 0

Author

Brooke Logan, Jun 24 2016

Keywords

• nonn

Links

• Kival Ngaokrajang, Illustration of initial terms

Crossrefs

If 4i in the definition is replaced by 2i we get A117571, and if 4i is replaced by 3i we get A273724.

Programs

Maple

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 ij then if h[j,i]=0 then ct:=ct+1; h[j,i]:=1; fi;
fi;
od:
ans:=[op(ans),ct];
od:
ans;  # N. J. A. Sloane, Jun 24 2016

Formula

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)).