cp's OEIS Frontend

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.

%I A274462 #26 Jun 25 2016 14:14:24
%S A274462 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,
%T A274462 24,27,28,21,30,31,30,33,32,33,36,37,36,37,40,39,42,43,36,45,46,45,48,
%U A274462 47,48,51,52,51,52,55,54,57,58,51
%N 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.
%H A274462 Kival Ngaokrajang, <a href="/A274462/a274462.pdf">Illustration of initial terms</a>
%F A274462 We argue as in A273724. There are n-1 choices for i.
%F A274462 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.
%F A274462 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.
%F A274462 Putting the pieces together, we obtain the g.f.
%F A274462 8 + x^2/(1-x)^2 - 2/(1-x^3) - 2(x^5+x^10)/(1-x^15) - 6/(1-x^15),
%F A274462 which can be rewritten as
%F A274462 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)).
%p A274462 M:=4; # M is the multiplier (2 for A117571, 3 for A273724, 4 for the present sequence)
%p A274462 ans:=[0,0];
%p A274462 for n from 2 to 100 do
%p A274462 h:=Array(0..n-1,0..n-1,0); ct:=0;
%p A274462 for i from 1 to n-1 do j := (M*i mod n);
%p A274462 if i<j then if h[i,j]=0 then ct:=ct+1; h[i,j]:=1; fi;
%p A274462 elif i>j then if h[j,i]=0 then ct:=ct+1; h[j,i]:=1; fi;
%p A274462 fi;
%p A274462 od:
%p A274462 ans:=[op(ans),ct];
%p A274462 od:
%p A274462 ans;  # _N. J. A. Sloane_, Jun 24 2016
%Y A274462 If 4i in the definition is replaced by 2i we get A117571, and if 4i is replaced by 3i we get A273724.
%K A274462 nonn
%O A274462 0,5
%A A274462 _Brooke Logan_, Jun 24 2016